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

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Solution

There are total three cases.
$(i) : 4$ Bowlers $+ 5$ Batsmen $+ 2$ WK
No. of ways $=^{6} C_{4} \cdot^{7} C_{5} \cdot^{2} C_{2} = 315$

$(i i) : 4$ Bowlers $+ 6$ Batsmen $+ 1$ WK
No. of ways $=^{6} C_{4} \cdot^{7} C_{6} \cdot^{2} C_{1} = 210$

$(i i i) : 5$ Bowlers $+ 5$ Batsmen $+ 1$ WK
No. of ways $=^{6} C_{5} \cdot^{7} C_{5} \cdot^{2} C_{1} = 252$

Total number of ways $= 315 + 210 + 252 = 777$

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