<s>[INST] A class consists of 15 boys of whom 5 are prefects. How many committees of 8 can be formed if each consists of at least 2 prefects? A class consists of $15$ boys of whom $5$ are prefects. How many committees of 8 can be formed if each consists of at least $2$ prefects?
I had the solution following solution in mind but that seems to be incorrect. can anyone please explain to me why?
First, I select $2$ students from the perfect category using $\binom{5}{2}$ and then I select remaining $6$ students combined from both the perfect and unperfect category i.e. $\binom{13}{6}$. So the answer becomes $\binom{5}{2}\times \binom{13}{6} = 17160$.
But, the correct answer is $5790$. Can you please help me?
 [/INST] The total number of committees of $8$ is $\binom{15}{8}$. The number of committees of $8$ with no prefects is equal to $\binom{10}{8}$. The number of committees with exactly one prefect is equal to $\binom{5}{1} \cdot \binom{10}{7}$.
Then the answer to your question is $\binom{15}{8} - \binom{10}{8} - \binom{5}{1} \cdot \binom{10}{7}=6435 - 45 - 5\cdot 120 = 5790.$
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