In how many ways can a cricket team of eleven players be chosen out of a batch…

Concept: When selecting r items from n items where the order of selection does not matter, the number of ways is given by the combination formula: nCr = n! / (r! × (n − r)!).

1. Here, n = 15 (total players) and r = 11 (players to be chosen for the team).

2. Since choosing 11 players to include is the same as choosing 15 − 11 = 4 players to leave out, 15C11 = 15C4 (using the symmetry nCr = nC(n−r)).

3. 15C4 = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1).

4. Numerator: 15 × 14 × 13 × 12 = 32760.

5. Denominator: 4! = 24.

6. 32760 / 24 = 1365.

Cross-check: Computing 15C11 directly via 15! / (11! × 4!) reduces to the same 15 × 14 × 13 × 12 / 4! expression, since the 11! in the denominator cancels with the corresponding descending terms of 15!, confirming the result independently.

So the cricket team of 11 players can be chosen from 15 players in 1365 ways.