Which is equal to 8C3?

8C5. Choosing 3 to take is the same as choosing 5 to leave.

In how many ways can a team of 2 be chosen from 6 people?

15. (6 × 5) ÷ 2: the order does not matter.

1. There is one way to choose nothing.

How is "at least one girl" usually counted?

the total, minus the number with no girls. It avoids adding up several separate cases.

Combinations and selections · Cambridge IGCSE Additional Mathematics

In plain words

Choosing three people for a team is different from choosing a first, second and third. In a team, the order does not matter: Ana, Ben and Cy is the same team as Cy, Ben and Ana. Counting selections is the subject of combinations.

6 things to know

A combination is a selection: the order does not matter.
The number of ways of choosing r objects from n is n! ÷ ((n − r)! × r!). It is written as nCr.
Choosing r to take is the same as choosing n − r to leave, so 8C3 = 8C5. Also nC0 = 1 and nCn = 1.
When choosing from two groups, multiply: 2 men from 5 and 3 women from 6 is 5C2 × 6C3.
When there are separate cases, such as "2 women or 3 women", work out each and add.
"At least one" is easiest as the total with no restriction, minus the number with none.

Worked example

A committee of 5 is chosen from 5 men and 6 women. In how many ways can it have exactly 2 men?

Choose the 2 men: 5C2 = 10.
The other 3 must be women: 6C3 = 20.
10 × 20 = 200 ways.

Worked example

A group of 3 is chosen from 4 men and 5 women. In how many ways does it include at least one woman?

With no restriction: 9C3 = 84.
With no women at all, all 3 are men: 4C3 = 4.
84 − 4 = 80 ways.

Tips and tricks

Ask "does the order matter?" A team or a committee: combinations. A queue or a code: permutations.
"And" means multiply. "Or" means add.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Combinations and selections: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is 8C3?
- 24
- 56 (the answer)
- 336
- 512
(8 × 7 × 6) ÷ 6.
Which is equal to 8C3?
- 8C2
- 8C4
- 8C5 (the answer)
- 3C8
Choosing 3 to take is the same as choosing 5 to leave.
In how many ways can a team of 2 be chosen from 6 people?
- 12
- 15 (the answer)
- 30
- 36
(6 × 5) ÷ 2: the order does not matter.
What is nC0?
- 0
- 1 (the answer)
- n
- undefined
There is one way to choose nothing.
How is "at least one girl" usually counted?
- the number with exactly one girl
- the total, minus the number with no girls (the answer)
- the number with all girls
- the total, plus the number with no girls
It avoids adding up several separate cases.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 8 marks. Write your answers on paper, then check them.