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Nov 2018 p61 q3
2828
In an orchestra, there are 11 violinists, 5 cellists and 4 double bass players. A small group of 6 musicians is to be selected from these 20.
How many different selections of 6 musicians can be made if there must be at least 4 violinists, at least 1 cellist and no more than 1 double bass player?
Solution
We need to consider different scenarios based on the given conditions:
4 Violinists (V), 1 Cellist (C), and 1 Double Bass player (DB):
The number of ways to choose 4 violinists from 11 is given by \(\binom{11}{4}\).
The number of ways to choose 1 cellist from 5 is given by \(\binom{5}{1}\).
The number of ways to choose 1 double bass player from 4 is given by \(\binom{4}{1}\).
Total ways for this scenario: \(\binom{11}{4} \times \binom{5}{1} \times \binom{4}{1} = 6600\).
4 Violinists and 2 Cellists:
The number of ways to choose 4 violinists from 11 is \(\binom{11}{4}\).
The number of ways to choose 2 cellists from 5 is \(\binom{5}{2}\).
Total ways for this scenario: \(\binom{11}{4} \times \binom{5}{2} = 3300\).
5 Violinists and 1 Cellist:
The number of ways to choose 5 violinists from 11 is \(\binom{11}{5}\).
The number of ways to choose 1 cellist from 5 is \(\binom{5}{1}\).
Total ways for this scenario: \(\binom{11}{5} \times \binom{5}{1} = 2310\).
Adding all scenarios together gives the total number of ways:
