Permutations and combinations in probability

Warm-up

Three quick questions from earlier lessons.

Discover

In how many orders can 3 books stand on a shelf?

6
3

Key idea

Count ordered choices; divide when order does not matter.

Choosing

r

things from

n

in order (a permutation):

_nP_r = n(n - 1) \cdots (n - r + 1)

. Choosing

r

things without order (a combination):

_nC_r = \dfrac{_nP_r}{r!}

, because each group was counted

r!

times. With equally likely outcomes, a probability is (favorable count) ÷ (total count).

Worked example

Choose 2 of 5 students: as president and treasurer, or as a pair.

5 \cdot 4 = 20

In order.

20 \div 2! = 10

As a pair.

Your turn, guided

Your turn: choose 2 of 6 for a committee.

15

6 · 5 ÷ 2.

Practice ★

5 runners are in a race. In how many ways can the gold, silver and bronze medals be awarded?
10 runners are in a race. In how many ways can the gold, silver and bronze medals be awarded?

Which question is answered by counting arrangements where order matters (a permutation)?
1. In how many ways can 5 students stand in a line for a photo?
2. In how many ways can 2 of the 5 students be chosen for a committee?
3. How many handshakes happen if each of 5 students shakes hands once with every other?

Watch out!

Practice ★★

In how many ways can a team of 3 students be chosen from 5 students?
In how many ways can a team of 3 students be chosen from 7 students?

Two students are chosen at random from a class of 4 to go on a trip. What is the probability that both Ana and Ben are chosen?

Exit ticket

Three questions to check you have got it.

5 runners are in a race. In how many ways can the gold, silver and bronze medals be awarded?
10 runners are in a race. In how many ways can the gold, silver and bronze medals be awarded?
6 runners are in a race. In how many ways can the gold, silver and bronze medals be awarded?

Summary card

Permutations and combinations

Order matters: nPr. Order does not: nCr = nPr ÷ r!.

Watch out: ask whether swapping two choices changes the result.

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