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Worked solution

Choose people while controlling complete couples

In each of the following cases, find the number of ways of forming a group of 10 people from 15 couples if each group contains

(i) exactly 5 couples, mark

(ii) no couples, marks

(iii) exactly 2 couples. marks

[2011/NYJC/II/7]

CJC · Review problems · Question 1

Answer

(i) 3003; (ii) 3075072; (iii) 11531520.

(i) Five complete couples

Choose the five couples: (155)=3003\binom{15}{5}=3003. No further choices are needed.

How did this part go?

(ii) No complete couples

Choose 10 of the 15 couples, then exactly one of the two partners from each:

(1510)210=3075072.\binom{15}{10}2^{10}=3075072.

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(iii) Exactly two complete couples

Choose two whole couples first. The other six people must come from six distinct couples among the remaining 13, with one of two partners from each:

(152)(136)26=11531520.\binom{15}{2}\binom{13}{6}2^6=11531520.

How did this part go?

Common mistakes

Choosing the remaining six people freely could accidentally add a third complete couple.

Solution notes

Every source selection factor and its explanatory role retained; source marks 1,2,3.

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