Permutations and Combinations

Permutations: A permutation is an arrangement of certain objects and thus the ordering of objects is important.

Factorial: The continued product of first 'n' natural numbers is called "n-factorial" and denoted by n!

n!= 1х2х3х.........(n-1)n : 0!=1, ⁿP₀=1, ⁿPₙ= n!

Factorial of negative number is undefined.

n!=n(n-1)!

(n-1)!=n!/n

if n=0 then (-1)!=0!/0 = 1/0 not defined.

ⁿPᵣ formula:

An r- permutations of 'A' can be formed in 'r' steps, as described below:

First, choose an object from 'A' put it in the first position. next choose an object from remaining ones in 'A' and put it in second position.

⬜ ⬜ ⬜..................... ⬜ ⬜

1st 2nd 3rd '(r-1)'th 'r' th

There are 'n' choices in 1st step, (n-1) choices in 2nd step,..............[n-(r-2)] choices in '(r-1)'th step and

[n-(r-1)] choices in 'r' th step.

Thus by (MP) multiplication principle

ⁿPᵣ = n(n-1)(n-2)................[n-(r-2)][n-(r-1)]

Multiply and divide with (n-r)[n-(r +1)].........2.1

ⁿPᵣ = n(n-1)(n-2)................[n-(r-2)][n-(r-1)](n-r)[n-(r +1)].........2.1 ⁄ (n-r)[n-(r +1)].........2.1

ⁿPᵣ= n!/(n-r)! {∵n!= 1х2х3х.........(n-1)n}

- Kumar.