# Permutations

## PERMUTATIONS

A permutation is an arrangement of elements in specific order.

Arranging numbers, letters, colors, books, objects, people, rank (1st, 2nd, 3rd place …) etc.

Example: In how many ways can the letters in A, B, C, be arranged?

The possible permutations are:

Hence, there are six distinct arrangements:

ABC
ACB
BAC
BCA
CAB
CBA

The order is important:      A B C differs from C B A

The number of ways is 3•2•1 =  6

## FACTORIAL

n! is the product of all positive integers less than or equal to n.

3! = 3•2•1 =  6

The number of different ways the three elements can be arranged is 3!

Examples:

1! = 1•1 = 1
2! = 1•2 = 2
3! = 1•2•3 = 6
4! = 1•2•3•4 = 24

0! = 1,  by definition

The number of permutations of n distinct elements, taken r at a time is:  P(n,r)

P(3,3) represents the number of permutations of 3 elements taken 3 at a time.

The general formula for P(n,r) is:

Example: How many 2-letter words can be made with the letters A, B, C if no letter can be repeated?

Solution:

There are 3 letters (A , B and C) that can be selected in the first event.

There are 2 letters that can selected in the second event. left (1 letter was already used)

Total = 3 • 2 = 6

P(3,2) represents the number of permutations of 3 elements taken 2 at a time.  P(3,2)=6

Example. How many ways can we select a president, secretary and treasurer from a group of 10 people?  Find P(10,3)

Procedure:

First, we need to determine how many choices there are for each place: There are ten places for the first position, 9 for the second, and 8 for the third.

Second, Find the product: 10•9•8=720

P(10,3) = 720

## COMBINATIONS

A combination is an arrangement in which order does not matter.

Example: There are three workers: Adam, Ben and Carl, for simplicity we call them the set {A, B, C}

How many two-man crews can be selected from this set?

Answer: There are three different two-man crews:

AB

AC

BC

The order did not matter since AB and BA represented the same two-man crew (also AC = CA and BC = CB).

We have three ways to choose the first element and two ways to choose the second (3•2=6). But since the order does not matter, we can discard some combinations. Therefore, we need to divide by the number of different ways the two elements can be arranged (2!)

The main difference in the definition of a permutation and a combination is whether the order is important.