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PERMUTATIONS AND COMBINATIONS SUMMARY

PERMUTATIONS AND COMBINATIONS SUMMARY FOR CA FOUNDATION EXAMS.

  • Fundamental principles of counting

(a) Multiplication Rule: If certain thing may be done in ‘m’ different ways and when it has been done, a second thing can be done in ‘n ‘ different ways then total number of ways of doing both things simultaneously = m × n.

(b) Addition Rule : It there are two alternative jobs which can be done in ‘m’ ways and in ‘n’ ways respectively then either of two jobs can be done in (m + n) ways.

  • Factorial: The factorial n, written as n! or ⌊n, represents the product of all integers from 1 to n both inclusive. To make the notation meaningful, when n = o, we define o! or ⌊o = 1.Thus, n! = n (n – 1) (n – 2) ….. …3.2.1
  • Permutations: The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or collection of objects with due regard being paid to the order of arrangement or selection, are called permutations.

The number of permutations of n things chosen r at a time is given by

nPr =n ( n – 1 ) ( n – 2 ) … ( n – r + 1 )
where the product has exactly r factors.

  • Circular Permutations: (a) n ordinary permutations equal one circular permutation. Hence there are nPn / n ways in which all the n things can be arranged in a circle. This equals (n–1)!.

(b) the number of necklaces formed with n beads of different colours = 1/2 (n-1)

  • (a) Number of permutations of n distinct objects taken r at a time when a particular object is not taken in any arrangement is n–1pr.

(b) Number of permutations of r objects out of n distinct objects when a particular object is always included in any arrangement is r. n-1Pr-1.

  • Combinations: The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important, are called combinations.

nCr = n!/r! ( n – r )!
nCr = nCn–r
nCo = n!/{0! (n–0)!} = n! / n! =1.
nCn = n!/{n! (n–n)!} = n! / n! . 0! = 1.

  • nCr has a meaning only when r and n are integers 0 ≤ r ≥ n and nCn–r has a meaning only when 0 ≤ n – r ≥ n.
    (i) n+1Cr = nCr + nCr–1
    (ii) nPr = n–1Pr +rn–1 Pr–1 
  • Permutations when some of the things are alike, taken all at a time

P = n!/ n1! n2! n3!

  • Permutations when each thing may be repeated once, twice,…upto r times in any arrangement = n!.
  • The total number of ways in which it is possible to form groups by taking some or all of n things (2n–1).
  • The total, number of ways in which it is possible to make groups by taking some or all out of n (=n1 + n2 + n3 +…) things, where n1 things are alike of one kind and so on, is given by

{ (n1 + 1) ( n2 + 1) ( n3 + 1)…} –1

  • The combinations of selecting r1 things from a set having n1 objects and r2 things from a set having n2 objects where combination of r1 things, r2 things are independent is given by

n1Cr1 x n2Cr2

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