CAIE A-Level Mathematics AS 5.3.1 Permutations and Combinations Questions

Practise calculating probabilities for arrangements, selections and repeated independent trials.

Exam points

calculate probabilities from equally likely outcomes
use complementary and conditional counting ideas in probability
model repeated trials and events using probability rules

[Maximum number: 4]

The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.

Find the probability that the three Rs are in the same group.

The favourable cases are:

- Group of 5 contains the three R's and two E's: 1 way.
- Group of 5 contains the three R's and one E: $\binom21\binom41=8$ ways.
- Group of 5 contains the three R's and no E: $\binom42=6$ ways.
- Group of 4 contains the three R's and one E: $\binom21=2$ ways.
- Group of 4 contains the three R's and no E: $\binom41=4$ ways.

Thus the total number of favourable splits is 21. The total number of splits is $\binom95=126$ , so
$P(\text{three }R\text{'s in the same group})=\frac{21}{126}=\frac16=0.167.$

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CAIE A-Level Mathematics AS 5.3.1 Permutations and Combinations Questions question 1