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