JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Permutations and Combinations

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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 ______.

Answer: 777

Solution

The minimum requirement of

4

bowlers,

5

batsmen and

1

wicketkeeper already fills

10

of the

11

places, so exactly one extra player is chosen and only three compositions are possible. Five bowlers, five batsmen and one wicketkeeper give

^{6} C_{5}^{7} C_{5}^{2} C_{1} = 6 \times 21 \times 2 = 252

teams. Four bowlers, six batsmen and one wicketkeeper give

^{6} C_{4}^{7} C_{6}^{2} C_{1} = 15 \times 7 \times 2 = 210

teams. Four bowlers, five batsmen and two wicketkeepers give

^{6} C_{4}^{7} C_{5}^{2} C_{2} = 15 \times 21 \times 1 = 315

teams. The three cases are mutually exclusive, so the number of selections is

252 + 210 + 315 = 777

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Integer answer

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JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Permutations and Combinations

Solution

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