# KS5. S1. Combinatorics

Permutations

How many ways can n objects be arranged in a line?

This number has a special name, “n factorial” and a special symbol “n!”. Find the button that calculates this number for you on the calculator.

How many ways can n objects be arranged in a line of length r?

This is n! / (n-r)! and can be calculated on the calculator using the nPr button.

Permutations (i.e. rearrangements) of n objects of which p of them are the same as each other, and q are the same as each other and r are the same as each other (etc.) can be calculated as $\frac{n!}{p!q!r!...}$

Worked Example 1

a. Find the number of ways in which all five letters in the word GREAT can be arranged.

b. In how many of these arrangements are the letters A and E next to each other?

Worked Solutions 1

Worked Example 2

Find the number of ways in which all five letters in the word GREET can be arranged.

Worked Example 3

How many different arrangements of the letters in the word MATHEMATICAL are there?

Exercises

Permutations with Restrictions

Restrictions will reduce the total number of arrangements possible. Problems are typically tackled by considering the restricted positions first and then the ones without restrictions.

Worked Examples

Line of Men Example

Find the number of ways of arranging six men in a line so that:

1. the oldest man is at the far-left side;
2. the two youngest men are at the far-right side;
3. the shortest man is at neither end of the line.

Four Digit Number Example

How many odd four-digit numbers greater than 3000 can be made from the digits 1, 2, 3 and 4, each used once?

Mangos and Watermelons Example

How many ways can 3 watermelons and 2 mangos be placed in a line if each individual piece of fruit is distinguishable and the mangos:

(a) must not be separated;

(b) must be separated.

Exercise

Permutations of r out of n objects

Reminder of theory:

Worked Examples

1. How many three-digit numbers can be made from the seven digits 3, 4, 5, 6, 7, 8 and 9, if each is used at most once?
2. In how many ways can five playing cards from a standard deck of 52 cards be arranged in a row?
3. In how many ways can 4 out of 18 girls sit on a four-seat sofa when the oldest girl must be given one of the seats?
4. In how many ways can four boys and three girls stand in a row when no two girls are allowed to stand next to each other?

Exercise

Combinations

Combinations are similar to permutations, i.e. they are a count of the different arrangements of objects, but with combinations the order doesn’t matter, i.e. 1,2,3 and 3,2,1 are the treated as the same arrangement. So of course there are less combinations of a group of objects than there are permutations of it.

An example of a situation modelled by a combination is the number of ways of selecting a committee of r people from a group of n people.

In any such situation we can use the formula: nCr = ${n \choose 4}$ = $\frac{n!}{(n-r)!r!}$  (find nCr button on calculator)

Worked Examples

Governor Example

A committee of five school governors is to be chosen from 8 applicants. How many different selections are possible?

Committee Example

In how many ways can a committee of four people be selected from four applicants?

Books and Magazines Example

In how many ways can five books and three magazines be selected from eight books and six magazines?

Exercise

Problem Solving with Permutations & Combinations

If we are able to spot a situation in a problem where the situation relates to a permutation or a combination, we can significantly reduced the amount of working needed to tackle the problem.

Worked Examples

Tins Example

There are 15 identical tins on a shelf. None of the tins are labelled but it is known that eight contain soup (S), four contain beans (B) and three contain peas (P).

If seven tins are randomly selected without replacement, find the probability that exactly five of them contain soup.

Cherries Example

A girl has a bag containing 13 red cherries (R) and seven black cherries (B). She takes five cherries from the bag at random. Find the probability that she takes more red cherries than black cherries.

Minibus Example

A minibus has seats for the driver (D) and seven passengers, as shown.
When seven passengers are seated in random order, find the probability that two particular passengers, A and B, are sitting on:

1. the same side of the minibus
2. opposite sides of the minibus.

Exercise

Mixed Exercise