JEE Main 2025 (Online) 7th April Morning ShiftQuestion 1
From a group of $7$ batsmen and $6$ bowlers, $10$ players are to be chosen for a team, which should include atleast $4$ batsmen and atleast $4$ bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is
Permutation and Combination
Solution
Concept. The captain (a batsman) and vice-captain (a bowler) are already in. Choose the other 8 from 6 batsmen and 5 bowlers. Formulas used. (1) $\binom nr$ ways to choose $r$ from $n$Step 1 - Conditions. Team needs $\ge4$ batsmen and $\ge4$ bowlers, so among the 8 extra players: $\ge3$ batsmen and $\ge3$ bowlers. Step 2 - Cases (extra batsmen, extra bowlers). $(3,5)$: $\binom63\binom55=20\cdot1=20$ $(4,4)$: $\binom64\binom54=15\cdot5=75$ $(5,3)$: $\binom65\binom53=6\cdot10=60$ Step 3 - Add. $20+75+60=155$. Answer: C