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Permutations and Combinations are widely used terms in Mathematics. A Permutation is a set of objects that is a listing of the objects in some specific order. The number of different permutations of n is given by n! (read as “n factorial”).

## Permutations:

A Permutation is an ordered combination of a set of objects. It is one of the different arrangements of a group of objects where order matters.

For example

3! = (3)(2)(1) = 6

4! = (4)(3)(2)(1) = 24

5! = (5)(4)(3)(2)(1) = 120

6! = (6)(5)(4)(3)(2)(1) = 720

The number of permutations of n objects taken r at a time is:
P(n,r) = n!/(n-r)!
Ex: Find the number of ways to arrange 4 people in groups of 3 at a time where order matters.

P(4,3) = 4!/(4-3)! = 24/1! = 24.

Therefore, there are 24 ways to arrange 4 items taken 3 at a time when order matters.

## Combinations:

A Combination on the other hand is a mix of different objects, with no order. A Combination is one of the different arrangements of a group of items where order does not matter.

The number of combinations of a group of n objects taken r at a time is:

C(n,r) = n!/r!(n-r)!

Ex: Find the number of ways to take 4 people and place them in groups of 3 at a time where order does not matter.

C(4,3) = 4!/3!(4-3)! = 24/6 = 4.

Therefore, there are 4 different ways of arranging 4 objects taken 3 at a time when order does not matter.

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