A committee of 6 is formed from 7 Indians and 4 Nepalese. In how many ways can this be done when the committee must contain at least two Nepalese?

Steps to Solve

1. Identify Committee Composition

The committee must contain 6 members with at least 2 Nepalese members. The possible compositions based on the number of Nepalese members are:

• 2 Nepalese and 4 Indians
• 3 Nepalese and 3 Indians
• 4 Nepalese and 2 Indians

2. Calculate Each Case

Now we will calculate the number of ways to form the committee for each case:

Case 1: 2 Nepalese and 4 Indians

• Choose 2 Nepalese from 4: $$ \binom{4}{2} = \frac{4!}{2!(4-2)!} = 6 $$
• Choose 4 Indians from 7: $$ \binom{7}{4} = \frac{7!}{4!(7-4)!} = 35 $$
• Total for Case 1: $$ 6 \times 35 = 210 $$

Case 2: 3 Nepalese and 3 Indians

• Choose 3 Nepalese from 4: $$ \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 $$
• Choose 3 Indians from 7: $$ \binom{7}{3} = \frac{7!}{3!(7-3)!} = 35 $$
• Total for Case 2: $$ 4 \times 35 = 140 $$

Case 3: 4 Nepalese and 2 Indians

• Choose 4 Nepalese from 4: $$ \binom{4}{4} = 1 $$
• Choose 2 Indians from 7: $$ \binom{7}{2} = \frac{7!}{2!(7-2)!} = 21 $$
• Total for Case 3: $$ 1 \times 21 = 21 $$

3. Sum All Cases

Now, we will sum the totals from all cases to find the total number of ways to form the committee:

Total ways = Case 1 + Case 2 + Case 3 $$ 210 + 140 + 21 = 371 $$