From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which sho…

Question

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively should be included. Then the total number of ways such a selection can be made, is:

Accepted Answer

With captain (batsman) and vice-captain (bowler) already fixed, we need 8 more players from the remaining 6 batsmen and 5 bowlers, with at least 3 batsmen and at least 3 bowlers among them (so total batsmen $\ge4$, bowlers $\ge4$):

, Batsmen, Bowlers, Ways |
---|---|---|
 3, 5, $\binom{6}{3}\binom{5}{5}=20$ |
 4, 4, $\binom{6}{4}\binom{5}{4}=75$ |
 5, 3, $\binom{6}{5}\binom{5}{3}=60$ |

$$\text{Total} = 20+75+60 = 155$$

Solutions · 1