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Introduction to Permutation

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Introduction to Permutation - Lesson Summary

 
A permutation is an arrangement in a definite order of a number of distinct objects taken some or all at a time.
 
Theorem on permutations
The theorem states that the number of permutations of n different objects taken r at a time, where 0 < r ≤ n and the objects do not repeat, is n(n - 1)(n - 2)...(n - r + 1), which is denoted by P r n .
 
If there are n objects and r spaces, these spaces need to be filled with n objects The first place can be filled using n objects in n ways. After filling the first place, the number of objects left is ‘n – 1’.

Now, the second place can be filled with ‘n -  1’ objects in ‘n - 1’ ways. Similarly, the third place can be filled in ‘n – 2’ ways. Continuing the same process, the r th place can be filled in ‘n - r + 1 ways.

According to the fundamental principle of multiplication,
total number of permutations = n × (n - 1) × (n - 2) × .... × (n - r + 1).
This expression is denoted by P r n .

P r n = n × (n - 1) × (n - 2) × .... × (n - r + 1), n ∈ N.
 


A convenient way to represent products such as n × (n - 1) × (n - 2) × .... × 2 × 1, 3 × 2 × 1 is a factorial.

Ex: 3 × 2 × 1 = 3!

As a general representation, n! = n × (n - 1) × (n - 2) × .... × 2 × 1.

4! = 4 × (4 – 1) × (4 – 2) × (4 – 3)

4! = 4 × 3 × 2 × 1 = 24.

Alternate representation of P r n .

P r n = n(n - 1)(n - 2)....(n - r + 1)

P r n = n(n - 1)(n - 2)....(n - r + 1) × (n - r)! (n - r)!  

P r n = n(n - 1)(n - 2)....(n - r + 1) × (n - r)(n - r - 1)....3 x 2 x 1 (n - r)!  

P r n = n(n - 1)(n - 2)....(n - r + 1) × (n - r - 1)(n - r)....3 x 2 x 1 (n - r)!  

P r n = n! (n - r)!  

This representation is used for computing P r n.

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