Combination – SS2 Further Mathematics Lesson Note (Second Term)

Combination is the number of ways r objects can be selected from n objects, irrespective of the arrangement. The order of selection does not matter. It is written nCr and found with the formula nCr = n! / ((n – r)! r!).

Behavioural objectives

By the end of the lesson, students should be able to:

• define combination and write its notation;
• state the formula for nCr;
• state the relationship between permutation and combination;
• evaluate nCr for given values of n and r;
• solve selection problems, including committee problems, using combination.

What is combination?

Combination can be defined as the number of ways r objects can be selected from n objects, irrespective of the arrangement.

Notation and formula

The notation is nCr or (n r), with n written above r in the bracket form.

nCr = n! / ((n – r)! r!)

Relationship between permutation and combination

nCr = nPr / r!

Worked examples

Example 1

Evaluate 10C4.

Solution:

10C4 = 10! / ((10 – 4)! 4!) = (10 x 9 x 8 x 7) / (4 x 3 x 2 x 1) = 10 x 3 x 7 = 210

Example 2

In how many ways can three books be selected from 12 books?

Solution:

n = 12, r = 3

12C3 = 12! / ((12 – 3)! 3!) = (12 x 11 x 10) / (3 x 2 x 1) = 220 ways

Example 3

A committee consisting of 3 men and 5 women is selected from 5 men and 10 women. Find how many ways this committee can be formed.

Solution:

Men: r = 3, n = 5

5C3 = 5! / ((5 – 3)! 3!) = 10

Women: r = 5, n = 10

10C5 = 10! / ((10 – 5)! 5!) = 252

Therefore, the number of ways of selecting the committee = 10 x 252 = 2520 ways.

Evaluation

1. Define combination.
2. Evaluate 7C3.
3. In how many ways can 4 pupils be selected from 9 pupils?
4. A committee of 2 men and 3 women is to be selected from 4 men and 6 women. In how many ways can this be done?

Assignment

1. Find the number of ways the letters of the word FURTHER can be arranged.
2. Find the number of ways of arranging 7 people in a straight line, if two particular people must always be separated.
3. In how many ways can 6 pupils be lined up if 3 of them insist on following one another?
4. Verify that [left side missing] = (n – 1)(n – 2)(n – 3)!