Legal Provisions relating to final Accounts

As per section 5(b): ‘Banking’ means the accepting for the purpose of lending or investment of deposits of money from public, repayable on demand or otherwise and withdrawable by cheque, draft, order or there wise.

  • As per section 5(d): ‘Company’ means any company defined in section 3 of the company’s act, 1956 and includes a foreign company within meaning of section 591 of that Act.
  • As per section 5(c): Banking Company means any company which transacts the business of banking in India.

Section 210 of the Companies Act governs the preparation of final account of a Company. The Board of Directors of a Company must, within 18 month from the date of incorporation, and subsequently once a year, lay take the company in general meeting the Balance Sheet of the Com­pany and a Profit and Loss Account.

In case of non-profit Companies, an Income and expenditure Account should be submitted. The period to which the account relates is called a Financial Year of the Company. It may be less or more of a calendar year but must not exceed 15 months. It may also be extended to 18 months provided the Register has granted special permission.

The Profit and Loss Account or Income and Expenditure Account the relate, in the case of the first Annual General Meeting of the Company, in a period from the date of a incorporation to a day which shall not precede the day of the meeting by more than nine months.

And in case of any subsequent Annual General Meeting, the period runs from the date of the previous accounts to a date not more than six months prior to the date of meeting.

Section 211 prescribes the form of Balance Sheet and contents of Profit and Loss Account. Every Balance Sheet of Company shall give a true and fair view of the state of affairs of the Company as at the end of the financial year.

It shall also be in the form set out in part 1 of Schedule VI, or in such other form as may be approved by the Central Government [211 (1)].

Provided that nothing contained in this sub-section shall apply to any Insurance or Banking Company or any Company engaged in the generation of supply of electricity or to any other class of Company for which a form of Balance Sheet been specified in or under the Act governing such class of Company.

Every Profit and Loss Account of a Company shall give a true and, fair view of the profit and loss the Company for the financial year. It shall also comply with the requirements of part II of Schedule VI, so far as they are applicable thereto [Sec. 211 (2)].

This requirement does not apply to any Insurance or Banking Company or any Company engaged in the generation of supply of electric­ity, or any other class of company for which a form of Profit and Loss Account has been specified in or under the Act governing such class of Company.

Every Balance Sheet and every Profit and Loss Account of a Company shall be duly signed on behalf of the Board of Directors by the Manager or Secretary, if any, and by not less than two Direc­tors of the Company.

One of the Directors who sign shall be a Managing Director where there is one. The Balance Sheet and Profit and Loss Account must be approved by the Board before they are submitted to the auditors who must in turn attach their report thereto.

The Profit and Loss Account shall be annexed to the Balance Sheet and the auditor’s report shall be attached thereto. If there is any separate, special supplementary report by the auditor, it shall also be attached to the Balance Sheet.

If any person, being a Director of a Company, fails to take all reasonable step to company with the previsions, he shall, in respect of each offence, be punishable with imprisonment for a term which may extend to six months, or with imprisonment for a term which may extend to six months, or with fine which may extend to Rs. 1,000 or with both.

However, the punishment of imprisonment is given only when the offence is committed willfully.

Profit and Loss Account:

The Indian Companies Act is silent as to the form of Profit and Loss Account. But part II of Schedule VI contains a list of items of incomes and expenditure which should be included in the Profit and Loss Account.

The profit and Loss Account of a Company should give a true and fair view of the profit or loss of the Company for the financial year. The first account covers the period since the incorporation of the Company, and subsequent accounts cover the period since the date of the preceding account.

An Income and Expenditure Account takes the place of Profit and Loss Account in the case of a Company not trading for profit.

Statutory Requirements:

Profit and Loss Account shall be so made out as to clearly disclose the result of the working of the Company during the period covered by the account and shall disclose every material feature including credits or receipts and debits or expenses in respect of non-recurring expenditure or expenditure of an exceptional nature. It shall set out the various items relating to the income and expenditure of the Company under the most convenient heads and in particular shall disclose the following information in respect of the accounting period.

The Profit and Loss Account must be prepared with the directions given in part II Schedule VI of the Act. The important provisions are given below:

  1. (a) The turnover, that is, the aggregate amount for which sales are effected by the Company, giving the amount of sales in respect of each class of goods dealt with by the Company, and indicating the quantities of such sales for each class Separately.

(b) Commission paid to sole selling agent within the meaning of section 294 of the Act

(c) Commission paid to other selling agents.

(d) Brokerage and discount on sales other than sales trade discount.

  1. (a) In the case of manufacturing concerns, the purchase of raw material including consumption and the opening and closing stocks of the good produced indicating the quantity produced.

(b) In the case of trading concerns the purchases made and the opening and the Closing stocks.

(c) In the case of Companies rendering or supplying services, the gross income derived from service rendered or supplied.

(d) In the case of Company which falls under more than one of the categories mentioned in (a), (b) and (c) above, it shall be sufficient compliance with the requirements herein if the total amounts are shown in respect of the opening and closing stocks, purchases, sales up and the gross income from services rendered is shown.

(e) In the case of other Companies, the gross income derived under different heads.

  1. In the case of all concerns having work-in-progress, the amounts for which (such works have been completed) at the commencement and at the end of the accounting period.
  2. The amount provided for depreciation, renewals or diminution in value of fixed assets. If such provision is not made by means of a depreciation charge, the method adopted for making such provision.

If not provision is made for depreciation, the fact that no provision has been made shall be stated and the quantum of arrears of depreciation computed in accordance with Section 205 (2) of the Act shall be disclosed by way of a note.

The amount of interest on the Company’s debentures and other fixed loans, that is to say loans for fixed periods stating separately the amount of interest if any, paid or payable to the Managing Director, and the Manager, if any.

The amount of charge for Indian Income-tax and other Indian taxation on profit and distinguishing them.

The amount reserved for:

(a) Repayment of share capital and

(b) Repayment of loans.

(a) The aggregate, if material, of any amounts set aside or proposed to be set aside to reserves but not including provisions made to meet any specific liability, contingency or commitment known to exist at the date of the Balance Sheet.

(b) The aggregate, if material, of any amount withdraws from such reserves.

(a) The aggregate, if material of the amounts set aside to provisions made for meeting specific liabilities, contingencies or commitments.

(c) The aggregate, if material, of the amounts withdrawn from such provisions, as no longer required.

Expenditure incurred on each of the following items separately for each item:

(a) Consumption of stores and spare parts.

(b) Power and fuel.

(c) Rent.

(d) Repairs to Buildings.

(e) Repairs to Machinery.

(f) (i) Salaries, Wages and Bonus.

(ii) Contribution to provident and other funds.

(iii) Workmen and staff welfare expenses to the extent not adjusted from any previous provision or reserve.

(g) Insurance

(h) Rates and taxes including taxes on income.

(i) Miscellaneous expenses.

Provided that any item under which the expenses exceed 1% of the total revenue of the Com­pany or Rs. 5,000, whichever is higher, shall be shown as a separate and distinct item against an appropriate account head in the Profit and Loss Account and shall not be combined with any other items to be shown under Miscellaneous Expenses.

(a) The amount of income from investments, distinguishing between trade Investments.

(b) Other income by way of interest, specifying the nature of the income.

(c) The amount of income tax deducted if the gross income is stated under subparagraphs (a) and (b) above.

(a) Profit and losses on investments (showing distinctly the extent of the profits or losses earned or incurred on account of membership of a partnership firm) to the extent not adjusted from any previous provision or reserve.

(b) Profit and losses in respect of transactions of a kind not usually undertaken by the Company or undertaken in circumstances of an exceptional or non-recurring if material in amount.

(c) Miscellaneous income.

(a) Dividends from Subsidiary Companies.

(b) Provisions for losses of Subsidiary Companies.

The aggregate amount of dividends paid and proposed, and stating whether such amounts are subject to deduction of income tax or not.

Amount, if material, by which any items shown in the Profit and Loss Account is affected by any change in the basis of accounting. The Profit and Loss Account shall contain by way of a note detailed information showing separately the following payments provided or made during the financial year to the Directors (in­cluding Managing Directors) the Managing Agents, Secretaries and Treasurers or Manager, if any, by the Company.

The subsidiaries the Company, the subsidiaries of the Company of any other person:

Managerial remuneration under Section 198 of the Act paid or payable during the financial year.

Expenses reimbursed to the Managing Agent under Section 345.

Commission or other remuneration payable separately to a Managing Agent or his associate under Section 356,357 and 358.

Commission received or receivable under Section 356 by the Managing Agent or his associate as selling agent of other concerns in respect of contracts entered into by such concerns with the Company.

The money value of the contracts for the sale or purchase of goods and materials or supply of services, entered into by the Company with the Managing Agent or his associate under Section 360 during the financial year.

Other allowances and commission including guarantee commission. Any other perquisites or benefits in cash or in kind stating the money value where practicable.

Pensions, gratuities, payments from provident funds in excess of own subscription and interest thereon, compensation for loss of office, consideration in connection with retirement from office.

The Profit and Loss Account shall contain or give by way of a note a statement showing the computation of net profits in accordance with Section 340 of the Act with the relevant details of the calculation of the commission’s payable by way of percentage of such profits to the directors including Managing Directors or Managers, if any.

The Profit and Loss Account also contains by way of note detailed information in regard to amounts paid to the auditors whether as fees, expenses or otherwise for services rendered as auditor and in any other capacity.

(VIII) Special points to be remembered while preparing Balance Sheet:

  1. Calls-in-arrears:

It refers to the amount not paid by the shareholders on the calls made on them by the company. This item is usually given in the trial balance. It should be deducted from the called up the liabilities side of the Balance Sheet to find paid up capital.

If the Trial balance shows only the paid up capital and the call-in-arrears is given in the adjustment, the amount is first added to the paid up capital to show the called up capital and then deducted against so that the paid up capital can be shown in the outer column.

  1. Unclaimed dividend:

It refers to the amount of dividend not collected by the shareholders from the company. This item is always shown on the credit side of Trial Balance. It is shown on the liabilities side of the Balance Sheet under the heading “Current Liabilities”.

  1. Forfeited shares account:

This item appears as a credit item in the Trial Balance and is shown on the liabilities side of the Balance sheet by adding it to the paid up capital.

  1. Securities premium account:

This item is shown on the liabilities side of the Balance Sheet under the heading “Reserves and Surplus”.

Rebate on Bills Discounted

Rebate on Bills Discounted is also known as Discount Received in Advance, or, Unexpired Discount or, Discount Received but not earned.

Its treatment is same as we do in the case of Interest Received in Advance.

Thus:

(i) If it is given only in the Trial Balance:

The same will be shown as a liability and will appear in the liability side of the Balance Sheet.

(ii) If it is given in adjustment:

In that case, the same is deducted from the Income from Interest and Discount in Profit and Loss Account and the same also will appear in the liability side of the Balance Sheet.

Method of Computation of Rebate on Bills Discounted:

For example, a customer discounts a bill of Rs. 60,000 for 3 months at 12% on 1st March 2000, it will be calculated as under:

Bank will earn discount @ 12% for 92 days i.e., = Rs. 60,000 x 12/100 x 92/365 = Rs. 1814.

But this amount of discount is meant for March, April and May. As accounts are prepared on 31st March each year, discount received for 61 days (30 + 31) for April and May is not actually earned. Thus, discount of 61 days i.e., Rs. 601 is called Rebate on Bill Discounted. So, actual income is Rs. 1213/- (i.e., 1814 – Rs. 601).

Treatment in Profit and Loss Account and Balance Sheet:

(i) Discount earned (up to 31.12.1992) will be credited to P & L A/c and unexpired discount will, however, be credited to Rebate on Bills Discounted A/c, and will appear in the liability side of the Balance Sheet.

(ii) Commissions earned will be credited to P & L A/c whereas customers’ liability for acceptance will appear on the asset side of the Balance Sheet and Acceptance on behalf of customer will appear in the liability side as contra items, and

(iii) Loans and advances will appear in the assets side of the Balance Sheet.

Slip system of Posting

It is a method of rapid posting in books maintained under Double Entry principle. Under this system, posting is done from slips and not from journals or cash books.

Slips are loose leaves of journals and these are supplied either by the customers or by the bank staff.

Reasons for adoption of slip system

  • Slip System of accounting/ posting is adopted for the following reasons:
  • It ensures smooth flow of accounting.
  • It can distribute the work of posting among many persons.
  • It helps in keeping the accounts of customer upto date.
  • This system saves a lot of clerical labour as most of the slips are filled in by the customers and provide an objective evidence of the records.

Features of slip system of posting

  • Slip system of posting is based on the Double Entry System. Double Entry System is a system in which each entry affects the two accounts with same amount, i.e. debit is equals to the credit.
  • The transactions are entered into by providing the details on the slips.
  • It contains the different slips i.e. withdrawal slip, pay-in-slip and cheque for different types of transactions.
  • It records the transactions in ledger book of accounts.
  • Slips serves as the evidence for the transactions.

Advantages of Slip System:

(i) it reduces the possibility of errors and frauds;

(ii) it saves a lot of time since it is prepared by the customers themselves;

(iii) it provides a good system of internal check etc.

Disadvantages of Slip System:

Chances of Loss of Slips: Slips are just like the loose leaves of the paper. It may be lost, distorted or misappropriated very easily. Thus, it is unjustified to maintain the records in form of the slips.

No Verification: As the posting is done directly to the ledger accounts and no entry is made in the subsidiary books. So verification of accounts is not possible.

Types

Withdrawal Slip

Withdrawal Slip is the slip used for drawing the amount from the bank account. It is a document of bank on which a person writes the date, account number and amount of money to withdraw from the bank. These slips contain the following particulars to be filled:

  • Name of the Drawer having bank account
  • Account Number
  • Date
  • Branch of the Bank
  • Signature of the Drawer

PAY-IN-SLIP

This is also known as deposit Slip. It is a form supplied by a bank for a depositor to fill out and to deposit the money in the bank. Pay-in-slip is used to deposit the amount in the bank. These slips contains the following particulars to be filled:

  • Name of the Depositor having bank account
  • Contact Number
  • Account Number
  • PAN Number
  • Date
  • Branch of the Bank
  • Denominations of the money deposited
  • Signature of the Depositor

CHEQUES

Cheque is a negotiable instrument which can be transferred by mere hand delivery. It is a document issued by an individual to his or her bank, directing them to pay the person whose name is mentioned in the document the sum specified in it for such a document to be valid, it is important that the person issuing it has an account in the said bank. An issuer of the cheque is called Drawer, and the one to whom it is issued is the Drawee. Cheque is used to make safe and convenient payment. It is less risky and danger of loss is minimized.

Acquisition of business when new set of books are opened

New set of books are opened

The following are the entries recorded by the purchasing company:

1. Purchasing consideration payable Business Purchase Account

  To Vendor Account

Dr.
2. Acquiring various assets and liabilities at agreed value Various Assets Account

   To Various Liabilities Account

   To Business Purchase Account

Dr.

Note: If the purchase price exceeds the net assets, the excess amount is debited to Goodwill Account; and if the net assets exceeds the purchase price, the excess amount is credited to Capital Reserve.

3. On payment of purchase price in kinds Vendor Account         Dr.
    To Share Capital Account
    To Debenture Account
    To Cash/Bank Account

Alternatively: The following entries can also Passed:
1. Acquiring various assets and liabilities at agreed value Various Assets Account Dr.
    To Various Liabilities Account  
    To Vendor Account  

 

Note: Any difference between the totals of debit and credit is debited to Goodwill Account or credited to Capital Reserve Account.

2. On payment of purchase price Vendor Account Dr.
    To Share Capital Account  
    To Debenture Account  
    To Cash/Bank Account  

Debtors and Creditors taken over on behalf of vendors

Sometimes the purchasing company does not take over the debtors and creditors of the vendor company but it agrees to collect book debts and pays off the creditors out of the collections from debtors. For this purpose, the company charges a commission by way of certain percentage both on the amount collected from debtors as well as the amount paid to creditors. Any profit or loss (in the form of discount and bad debts) should be borne by the vendor since the purchasing company does this work on behalf of the vendors.

The following entries are to be passed:

  1. Entries in the Books of Vendors:
(a) For Closing Debtors Account  
  Purchasing Company Suspense, A/c Dr.
      To Debtors A/c  

When same set of books are continued

But if the company desires to continue with the same set of books, the following accounting treatment should be suggested for the purpose:

(i) Open Revaluation Account:

A Revaluation Account should be opened where increased and decreased values of assets and liabilities are to be adjusted as we have seen in case of an admission of a partner. The profit or loss so made should be transferred to capital accounts.

(ii) Close Reserve:

Any balance of accumulated or undistributed profits or reserves should be transferred to capital accounts in profit-sharing ratio.

(iii) Close assets and liabilities not taken over:

Any asset or liability which is not taken over by the purchasing company should be transferred to capital accounts in profit- sharing ratio. But when an asset is worth its book value, the same can be transferred to capital accounts according to the ratio of final claim till the value of the asset is fluctuating.

(iv) Close Capital Accounts:

The Capital Accounts are then closed by transferring to Share Capital Amount, or Debentures Account or Cash Account.

Contributory Preferential Payments

The person appointed for conducting the liquidation proceedings of the company is called ‘Liquidator’. (In case of Voluntary winding up an Insolvency Professional). The company must submit a statement of affairs to the liquidator. The general duties of the liquidator are to take into his custody all the property of the company and actionable claims and make the payments as per the order laid down in the Companies Act.

Preferential payments: Preferential creditors are those creditors who are paid in priority to creditors having a floating charge and other (non-preferential) unsecured creditors. As per Sec. 326 of the Companies Act, 2013, preferential creditors include the following:

  1. All revenues, taxes, cesses and rates due to the Central, State Government or to a local authority which have become due and payable within twelve months before the date of winding up order.
  2. All wages or salaries of any employee not exceeding ` 20,000 per claimant, in respect of services rendered to the company and due for a period not exceeding four months within the said twelve months before the date of winding up order.
  3. All amounts due in respect of contribution payable during the twelve months under the Employees’ State Insurance Act, 1948 or any other law.
  4. Compensation due under Workmen’s Compensation Act, 1923 in respect of death or disablement of any employee of the company.
  5. Any amount due to any employee from provident fund, pension fund, gratuity fund for the welfare of the employees maintained by the company.
  6. Accrued holiday remuneration becoming payable to the employee or in case of his death, to any other person in his right, on termination of his employment before, or by the effect of the winding up.
  7. The expenses of any investigation held in pursuance of Sec. 213 or 216 in so far as they are payable by the company.

Overriding Preferential Payments (Section 326)

Overriding preferential payments are to be paid in priority to all other debts as per the said Act.

They include:

(a) Dues to workmen, and

(b) Debts due to secured creditors to the extent such debts rank to the security of every creditor shall be deemed to be subject to pari passu charge in favor of the workmen to the extent of workmen’s portion therein.

In a winding up, subject to the provisions of section 326, there shall be paid in priority to all other debts:

(a) all revenues, taxes, cesses and rates due from the company to the Central Government or a State Government or to a local authority at the relevant date, and having become due and payable within the twelve months immediately before that date;

(b) all wages or salary including wages payable for time or piece work and salary earned wholly or in part by way of commission of any employee in respect of services rendered to the company and due for a period not exceeding four months within the twelve months immediately before the relevant date, subject to the condition that the amount payable under this clause to any workman shall not exceed such amount as may be notified;

(c) all accrued holiday remuneration becoming payable to any employee, or in the case of his death, to any other person claiming under him, on the termination of his employment before, or by the winding up order, or, as the case may be, the dissolution of the company;

(d) unless the company is being wound up voluntarily merely for the purposes of reconstruction or amalgamation with another company, all amount due in respect of contributions payable during the period of twelve months immediately before the relevant date by the company as the employer of persons under the Employees’ State Insurance Act, 1948 or any other law for the time being in force;

(e) unless the company has, at the commencement of winding up, under such a contract with any insurer as is mentioned in section 14 of the Workmen’s Compensation Act, 1923, rights capable of being transferred to and vested in the workmen, all amount due in respect of any compensation or liability for compensation under the said Act in respect of the death or disablement of any employee of the company:

Provided that where any compensation under the said Act is a weekly payment, the amount payable under this clause shall be taken to be the amount of the lump sum for which such weekly payment could, if redeemable, be redeemed, if the employer has made an application under that Act;

(f) all sums due to any employee from the provident fund, the pension fund, the gratuity fund or any other fund for the welfare of the employees, maintained by the company; and

(g) the expenses of any investigation held in pursuance of sections 213 and 216, in so far as they are payable by the company.

(2) Where any payment has been made to any employee of a company on account of wages or salary or accrued holiday remuneration, himself or, in the case of his death, to any other person claiming through him, out of money advanced by some person for that purpose, the person by whom the money was advanced shall, in a winding up, have a right of priority in respect of the money so advanced and paid-up to the amount by which the sum in respect of which the employee or other person in his right would have been entitled to priority in the winding up has been reduced by reason of the payment having been made.

(3) The debts enumerated in this section shall:

(a) rank equally among themselves and be paid in full, unless the assets are insufficient to meet them, in which case they shall abate in equal proportions; and

(b) So far as the assets of the company available for payment to general creditors are insufficient to meet them, have priority over the claims of holders of debentures under any floating charge created by the company, and be paid accordingly out of any property comprised in or subject to that charge.

(4) Subject to the retention of such sums as may be necessary for the costs and expenses of the winding up, the debts under this section shall be discharged forthwith so far as the assets are sufficient to meet them, and in the case of the debts to which priority is given under clause (d) of sub-section (1), formal proof thereof shall not be required except in so far as may be otherwise prescribed.

(5) In the event of a landlord or other person distraining or having distrained on any goods or effects of the company within three months immediately before the date of a winding up order, the debts to which priority is given under this section shall be a first charge on the goods or effects so distrained on or the proceeds of the sale thereof:

  • Provided that, in respect of any money paid under any such charge, the landlord
  • Other person shall have the same rights of priority as the person to whom the payment is made.

(6) Any remuneration in respect of a period of holiday or of absence from work on medical grounds through sickness or other good cause shall be deemed to be wages in respect of services rendered to the company during that period.

Statements of Affairs

According to Sec. 454, within 21 days of the date of the winding-up order to the appointment of the official liquidator as provisional liquidator, the company has to submit a statement to the official liquidator as to the affairs of the company unless the Court otherwise orders. The statement must be in the prescribed form.

Procedure of Preparation of Statement of Affairs

For the preparation of Statement of Affairs, the following points are to be followed:

  1. First of all, take all assets which are not specifically pledged. These assets are taken at their realisable values. It may be noted that calls in arrears are also treated as an asset not specifically pledged to the extent of estimated realisable amount, but uncalled capital is not shown as an asset.
  2. Add to the realisable value of the assets not specifically pledged, any surplus from assets specifically pledged.
  3. From the total as obtained by adding (1) and (2) first deduct the amount of preferential creditors, then the amount of creditors having a floating charge (e.g., debentures) and the result will be surplus or deficiency as regards debenture holders.
  4. Deduct the amount of unsecured creditors from the figure as obtained in (3) above; the resultant figure will be either surplus or deficiency as regards unsecured creditors.
  5. Deduct the amount of paid-up share capital to the figure as obtained in (4) above; the result will be either surplus or deficiency as regards members or contributories.
  6. Any unrecorded assets or liability should be shown both in the Statement of Affairs and the Deficiency or Surplus Account to make double entry complete.

Verified by affidavit and must contain the following particulars:

(i) The assets of the company, stating separately the cash in hand and cash at bank and negotiable securities.

(ii) The debts and liabilities of the company;

(iii) Names and addresses of its creditors, stating separately the amount of secured and unsecured debts;

(iv) In the case of secured debts, particularly of the securities held by the creditors, their value and dates on which they were given;

(v) The debts due to the company and names and addresses of the persons from whom they are due and the amount likely to be realized;

(vi) Such further information as may be required by the official liquidator.

Lists to be Attached to the Statement of Affairs

Following lists are attached to the Statement of Affairs:

  • List A gives a complete list of assets not specifically pledged in favour of secured creditors. Creditors having a floating charge on the assets are considered as having assets not specifically pledged with them; so such assets are included in the list.
  • List B gives the list of assets which are specifically pledged in favour of fully secured and partly secured creditors.
  • List C gives the list of preferential creditors.
  • List D gives the list of debenture holders and other creditors having a floating charge on the assets.
  • List E gives the names, addresses and occupations of unsecured creditors and the amount due.
  • List F gives the names and number and value of shares held by various preference shareholders.
  • List G gives the names and holdings of equity shareholders.
  • List H shows how Deficiency or Surplus in the Statement of Affairs has been arrived at, i.e., it explains the reasons responsible for the surplus or deficiency. According to the law, the period covered by Deficiency or Surplus must commence on a date not less than 3 years before the winding up order, or if the company has not been incorporated for the whole of that period, the date of incorporation of the company, unless the official Liquidator otherwise agrees.

Permutation, Combination

Permutation

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word “permutation” also refers to the act or process of changing the linear order of an ordered set.

Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.

Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.

The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.

Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3,1,2) mentioned above is described by the function alpha defined as:

α (1) = 3,

α (2) = 1

α (3) = 2

The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition (performing two given rearrangements in succession), which results in another rearrangement. As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set {1,2,…n} that are considered for studying permutations.

In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

Combination

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order.

Combinations can be confused with permutations. However, in permutations, the order of the selected items is essential. For example, the arrangements ab and ba are equal in combinations (considered as one arrangement), while in permutations, the arrangements are different.

Combinations are studied in combinatorics but are also used in different disciplines, including mathematics and finance.

Basics of Set Theory

Set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.

Fundamental set concepts

In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set may be defined by a membership rule (formula) or by listing its members within braces. For example, the set given by the rule “prime numbers less than 10” can also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers ℕ goes on forever. The empty (or void, or null) set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the status of being a set.

A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members of A are also members of B. For example, any set is a subset of itself, and Ø is a subset of any set. If both A ⊆ B and B ⊆ A, then A and B have exactly the same members. Part of the set concept is that in this case A = B; that is, A and B are the same set.

Representation of Sets

Sets can be represented in two ways:

  • Roster Form or Tabular form
  • Set Builder Form

Roster Form

In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }.

Example: If set represents all the leap years between the year 1995 and 2015, then it would be described using Roster form as:

A = {1996,2000,2004,2008,2012}

Also, multiplicity is ignored while representing the sets. e.g. If L represents a set that contains all the letters in the word ADDRESS, the proper Roster form representation would be

L = {A, D, R, E, S }= {S,E,D,A,R} 

L≠ {A, D, D, R, E, S, S}

Set Builder Form

In set builder form, all the elements have a common property. This property is not applicable to the objects that do not belong to the set.

Example: If set S has all the elements which are even prime numbers, it is represented as:

S= { x: x is an even prime number}

where ‘x’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

So, S = { x:x is an even prime number } is read as ‘the set of all x such that x is an even prime number’. The roster form for this set S would be S = 2. This set contains only one element. Such sets are called singleton/unit sets.

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