Question:
There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is
Solution:
15 : Players
6: Bowlers
7: Batsman
2 : Wicket keepers
Total number of ways for :
at least 4 bowlers, 5 batsman \& 1 wicket keeper
$={ }^{6} \mathrm{C}_{4}\left({ }^{7} \mathrm{C}_{6} \times{ }^{2} \mathrm{C}_{1}+{ }^{7} \mathrm{C}_{5} \times{ }^{2} \mathrm{C}_{2}\right)+{ }^{6} \mathrm{C}_{5} \times{ }^{7} \mathrm{C}_{5} \times{ }^{2} \mathrm{C}_{1}$
$=777$
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