A basket contains 3 mangoes, 4 apples, and 4 bananas. In how many ways can we…

When a selection must consist entirely of items from one group among several distinct groups, the number of ways to make that selection from a given group is the combination nCr = n! / (r! × (n − r)!), where n is the number of items available in that group and r is the number chosen. Since a selection here can be all mangoes, OR all apples, OR all bananas (three mutually exclusive cases), the total number of ways is the SUM of the combination count from every group that has at least r items.

1. Mangoes: 3 available, choose all 3 → 3C3 = 1 way.

2. Apples: 4 available, choose 3 → 4C3 = 4 ways.

3. Bananas: 4 available, choose 3 → 4C3 = 4 ways.

Total ways = 1 + 4 + 4 = 9 ways.

Cross-check using the symmetry identity nCr = nCn-r: 4C3 = 4C1 = 4 (choosing 3 out of 4 leaves 1 out, the same count), confirming the apple and banana terms; and 3C3 = 1 trivially, since there is exactly one way to choose all 3 items when only 3 are available. Adding these confirmed terms again gives 9, matching the '9 ways' option.