Question

(a) Five friends are in a netball squad. In each game during the 21-round season, at least 3 of them are picked in the team. Prove that there will be at least 3 matches in which the same three friends are selected to play.

(b) How does the answer change if there are six friends instead of 5?
PLS ANSWER FAST!!!!

Answer 1

(a) there are 10 sets of 3 friends, so in 21 games, at least one set must show 3 times

(b) there are 20 sets of 3 friends, so in 21 games, at least one set must show 2 times.

Explanation:

(a) The number of combinations of 5 things taken 3 at a time is ...

5C3 = 5!/(3!·2!) = 5·4/2 = 10

There can be 10 games in which the same 3 friends do not show up. There can be 10 more games such that the same 3 friends show up exactly twice. In the 21st game, some set of 3 friends must show up 3 times.

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(b) The number of combinations of 6 things taken 3 at a time is ...

6C3 = 6!/(3!·3!) = 6·5·4/(3·2) = 20

Hence, there can be 20 games in which the same 3 friends do not show up. In the 21st game, some set of 3 friends will show up a second time.