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

^{n}P_{r} =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
^{n}P_{n}/ 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–1}p_{r}.

(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-1}P_{r-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.

^{n}C_{r} = n!/r! ( n – r )!

^{n}C_{r} = ^{n}C_{n–r}

^{n}C_{o} = n!/{0! (n–0)!} = n! / n! =1.

^{n}C_{n} = n!/{n! (n–n)!} = n! / n! . 0! = 1.

^{n}C_{r}has a meaning only when r and n are integers 0 ≤ r ≥ n and^{n}C_{n–r}has a meaning only when 0 ≤ n – r ≥ n.

(i)^{n+1}C_{r}=^{n}C_{r}+^{n}C_{r–1}

(ii)^{n}P_{r}=^{n–1}P_{r}+r^{n–1}P_{r–1 }- Permutations when some of the things are alike, taken all at a time

P = n!/ n_{1}! n_{2}! n_{3}!

- 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 (2
^{n}–1). - The total, number of ways in which it is possible to make groups by taking some or all out of n (=n
_{1}+ n_{2}+ n_{3}+…) things, where n_{1}things are alike of one kind and so on, is given by

{ (n_{1} + 1) ( n_{2} + 1) ( n_{3} + 1)…} –1

- The combinations of selecting r
_{1}things from a set having n_{1}objects and r_{2}things from a set having n_{2}objects where combination of r_{1}things, r_{2}things are independent is given by

^{n1}C_{r1 }x ^{n2}C_{r2}