This section covers permutations and combinations.

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**Arranging Objects**

The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

**Example**

How many different ways can the letters P, Q, R, S be arranged?

The answer is 4! = 24.

This is because there are four spaces to be filled: _, _, _, _

The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!

The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is:

n! .p! q! r! …

**Example**

In how many ways can the letters in the word: STATISTICS be arranged?

There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are:

10!=50 4003! 2! 3!

**Rings and Roundabouts**

When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!

**Example**

Ten people go to a party. How many different ways can they be seated?

Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440

**Combinations**

The number of ways of selecting r objects from n unlike objects is:

**Example**

There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls?

10C3 =10!=10 × 9 × 8= 120 3! (10 – 3)!3 × 2 × 1

**Permutations**

A permutation is an ordered arrangement.

The number of ordered arrangements of r objects taken from n unlike objects is:

nPr = n! . (n – r)!

**Example**

In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.

10P3 =10! 7!

= 720

There are therefore 720 different ways of picking the top three goals.

**Probability**

The above facts can be used to help solve problems in probability.

**Example**

In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?

The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 .

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Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.