Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is ____________

Solution

Maths sir · 3 steps

Only three ways to split 5 fruits under these limits, so list the cases and add them up.

1. Find the splits. Let the counts be oranges O ≥ 2, red R ≥ 1 and white W ≥ 1, with O + R + W = 5. The minimums already use 4 fruits, so the one fruit left over can be an orange, a red apple or a white apple.
2. Count each split. Treat fruits of the same colour as different from one another. For each split, choose the oranges from 8, the red apples from 7 and the white apples from 5, then multiply.

   (O, R, W)	Ways	Value
   (3, 1, 1)	C(8, 3)C(7, 1)C(5, 1) = 56 × 7 × 5	1960
   (2, 2, 1)	C(8, 2)C(7, 2)C(5, 1) = 28 × 21 × 5	2940
   (2, 1, 2)	C(8, 2)C(7, 1)C(5, 2) = 28 × 7 × 10	1960

3. So. Total = 1960 + 2940 + 1960 = 6860.

The common slip: 17640. Fixing 2 oranges, 1 red and 1 white first and then choosing the 5th fruit from the remaining 18 gives 28 × 7 × 5 × 18 = 17640. That counts the same selection several times.

The idea to keep. With 'at least' conditions on several groups, list every allowed split and add the products. Fixing the minimums first and choosing the rest freely counts selections more than once.

Answer from the NTA final answer key. Solution by PrepWiser.