Permutation:

“Permutation means ARRANGEMENT”

Each of the different arrangement which can be made by taking some or all of a number or things is called a Permutation.

 Note: The word arrangement is used, if the order of things is considered.

Mathematically, The number of ways of arranging ‘n’ distinct objects , taking ‘r’at a time .

It is denoted by: ⁿP_r

ⁿP_r  =  n (n-1)(n-2)(n-3)……(n-r+1)

ⁿP_r   = n! / (n-r)!

 Solved Examples:

Q1.    Find ⁷⁵P₂   = ?

Ans: ⁿP_r =   ⁷⁵P_{2 = 75!/ (75-2)!} =75!/73!  = (75x 74x 73!) / 73! =                         75x 74 =5550

Q2.   Find ⁴P_{4  =?}    

Ans: ⁿP_r =  ⁴P_{4  =} 4!/ (4-4)! = 4!/0! = 4!/1 = 4! =4x 3x 2x 1 = 24.

Q3. How many different signals can be made by using  5 flags from 8-flags of different colours?

Ans.    Number of ways taking 5 flags out of 8-flag:  n = 8 and r = 5.

           ⁿP_r = ⁸P₅ = 8!/ (8-5)!  =  8! /3! = (8 x 7 x 6 x 5 x 4x 3!)/ 3! = 6720.

Q4.How many words can be made by using the letters of the word “SIMPLETON” taken all at a time?

Ans.   There are ‘9’ different letters of the word “SIMPLETON”

          n = n and r = 9.

          Number of Permutations taking all the letters at a time

           ⁿP_r = ⁹P₉   =  9!/ (9-9)! = 9!/0! = 9!  = 362880.

Q5.How many words with or without meaning can be formed by using all the letters of the word, PUNJAB’

Ans: Required number of words = ⁶P₆ 

          ⁿP_r =⁶P_{6 = 6!/ (6-6)!} = 6!/ 0! = 6!/1 = 6x 5 x 4 x 3 x 2 x1 = 720.

Q6. In how many ways can the letters of the word ‘LOVELY’ be arranged?

Ans: The word ‘LOVELY’ contains 6 letters:

L = 2 times, O = 1 time, V= 1 time, E = 1 time and Y= 1 time.

Required number of words = 6!/ (2! X 1! X 1! X 1! X 1!)

(Please note, 2! In denominator, because L  is repeated 2 times  )