Home | Math | Permutation and Combination – Definition, Formulae, and Examples

# Permutation and Combination – Definition, Formulae, and Examples

July 4, 2022
written by Azhar Ejaz

Permutation and Combination are used for lists (order necessary) or groups( order does not matter).

A permutation is used for lists and Combination is used for groups.

## Permutation

An arrangement of a finite number of objects taken some or all at a time is called a permutation. For example the permutation of n different objects taking r (where r â‰¤ n) at a time is denoted by nPr or P(n,r)

nPr = n!/(n-r)!

## Theorem:

nPr = n (n-1)(n-2)â€¦(n-r+1) = n!/(n-r)!

Proof:

Consider we have r places to fill n different objects. First place can be filled-in n different ways. When first place has been filled we are left (n-1) objects. Now second place can be filled in(n-1) different ways. First two places can be filled in n(n-1) ways.

After filling the second place we are left (n-2) different objects. Now third place can be filled in (n-3) different ways. First three places can be filled in n(n-1)(n-2) ways. Proceeding in this way by using fundamental law of counting, total number of arrangements in which r place can be filled up is given by

nPr = n(n-1)(n-2)(n-3)â€¦r factors = n(n-1)(n-1)(n-3)â€¦[n-(r-1)]

nPr = n(n-1)(n-2)(n-3)â€¦(n-r+1)

Multiply and divide R.H.S by (n-r)(n-r-1)â€¦3.2.1

nPr = n(n-1)(n-2)(n-3) x (n-r)(n-r-1)â€¦3.2.1/ (n-r)(n-r-1)â€¦3.2.1

nPr = n(n-1)â€¦(n-r+1)(n-r)(n-r-1)â€¦3.2.1/(n-r)(n-r-1)â€¦3.2.1

nPr  = n!/(n-r)!

Examples:

16P4 = 16!/(16-4)! = 16!/12! = 16.15.14.13.12!/12! = 16.15.14.13 = 43,680

14P3 = 14!/(14-3)! = 14!/11!

= 14.13.12.11!/11! = 14.13.12

= 2,184

## Permutation with related objects

### Theorem:

The number of permutations of n objects taken all at a time when one kind is q alike, the second kind is r alike, and third is s alike is given by

Proof:

Arrangements of q like objects = qPq = q!

Arrangement of s like objects = sPs = s!

Let x be the required number of permutations of n objects

Then total permutations = x. q!r!s!

But a number of permutations of n objects = n!

px. q!r!s! = n! => x = n!/q!r!s!

or

Example:

ASSESSEMENT

S is repeated 4 times

E is repeated 3 times

A,M,N and T come only 1 time.

## Combination

When the selection of objects is done neglecting its order, this is called combination.

Combinations of n different objects taken r at a time is denoted by nCr and defines as:

nCr = nPr/r!  or   nCr = n!/r!(n-r)!               nPr = n!/(n-r)!

For r = n        nCn = n!/n!(n-n)! = n!/n!0! = 1

For r = 0      nC0 = n!/0!(n-0)! = n!/0!n! = 1

## Prove that nCr = n!/r!(n-r)!

Proof

There are nCr combinations of n different objects taken r at a time.

Each combination consists of r different objects which can be permuted among themselves in r! permutation. Total number of permutation of n objects taken r at a time

nCr x r! = nPr => nCr = nPr/r! or nCr = n!/r!(n-r)!

## Difference between permutation and combination

• An arrangement where the order is important is called a permutation.
• An arrangement where the order is not important is called combination.

Examples:

10C3 = 10!/3!(10-3)! = 10!/3!7!

= 10.9.8.7!/3.2.1.7! = 120