Permutation and Combination: Permutation


Fundamental Counting Principle: If one work can be done by "a" different styles and a second work can be done by "b" different styles then the total syles of doing both works will be \((a \times b)\).

Factorial: The factorial of an integer is the product of all less than or equal positive integers. It is denoted by \(n!\), where n is any positive integer. The factorial of a negative integer is not possible.$$ n! = n \ (n - 1) \ (n - 2)......3 \times 2 \times 1 $$ $$ OR $$ $$ n! = 1 \times 2 \times 3.........(n - 1) \times n $$ $$ n! = (n - 1)! \times n $$

Example: \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Example: \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\) \(= 720\)

Note: Remember, \(0! = 1\), \(1! = 1\), \(nP_0 = 1\), \(nP_n = n!\).

Permutation: The selection of objects in an ordered manner is called a permutation.

Example: Three pens A, B, and C can be written in six different orders ABC, ACB, BAC, BCA, CAB, CBA Hence here the total number of permutation are 6.

Notation of Permutation: If there are "n" distinct objects and we select "r" objects together then the number of permutations is denoted as \(nP_r\) or P(n, r). $$ nP_r = n \ (n - 1) \ (n - 2)......r $$ $$ nP_r = \frac{n!}{(n - r)!} $$ Here (r < n).

Where, n = Total number of objects.
P = Permutation.
r = The number of selected objects at a time from the total number of distinct objects.

Important Formulae:

(i): n = total number of objects.

r = Number of selected objects at a time from the total number of distict objects. $$ nP_r = \frac{n!}{(n - r)!} $$

(ii): n = total number of objects.

p, q, r = the number of different selected objects at a time from the total number of distinct objects. $$ P(p, q, r) = \frac{n!}{p! q! r!} $$

Example: Find the value of \(10P_2\)?

Solution: $$ nP_r = \frac{n!}{(n - r)!} $$ $$ 10P_2 = \frac{10!}{(10 - 2)!} $$ $$ = \frac{10!}{8!} $$ $$ = \frac{10 \times 9 \times 8!}{8!} $$ $$ 10P_2 = 90 $$




Permutation and Combination

Chapter 1: Permutation


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