A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :

Solution

Maths sir · 3 steps

Foreigners must be twice the Indians, and only 8 exist, so list the cases with 2, 3 and 4 Indians.

1. Find the cases. Let the committee have i Indians, so it has 2i foreigners. We need i ≥ 2, and 2i ≤ 8 gives i ≤ 4. So i can only be 2, 3 or 4.

   Indians	Foreigners	Ways
   2	4	C(6, 2)C(8, 4) = 15 × 70 = 1050
   3	6	C(6, 3)C(8, 6) = 20 × 28 = 560
   4	8	C(6, 4)C(8, 8) = 15 × 1 = 15

2. Add the cases. The cases cannot overlap, because each one has a different number of Indians. So the counts add: 1050 + 560 + 15 = 1625.
3. So. The committee can be formed in 1625 ways, which is option (d).

The idea to keep. When a selection has a condition like at least 2, split it into separate cases, count each with a product of combinations, and add. The supply limit (here 8 foreigners) fixes the last case.

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