This is the question,
A student council has $10$ members. From this one President, one Vice-President, one Secretary, one Joint-Secretary and two Executive Committee members have to be elected. In how many ways this can be done?
- $151200$
- $75600$
- $37800$
- $18900$
And this is the answer on almost every website
Given, A student council has $10$ members. From this one President, one
Vice-President, one Secretary, one Joint, one Secretary and two Executive
Committee members have to be elected.$\implies
6$ members are elected out of $10$ members. The number of ways to
elect $6$ members out of $10$ members = $P(10,6) = 151200$.
This seems wrong to me. I think the answer should be like,
Out of $10$ we first select $1$ president, then out of $9$ one vice president, out of $8$ one Secretary, out of $7$ one joint Secretary and then out of $6$ two Executive Committee members.
This give us $\binom {10}{1} \times \binom {9}{1} \times \binom {8}{1} \times \binom {7}{1} \times \binom {6}{2} = 75600$.
Am I wrong somewhere?
Best Answer
Since an answer affirming the book answer has been put, I disagree with it.
Nowhere is it mentioned that particular seats are associated with the two Executive Committee members, your answer is correct, let us confirm it in a skightly different way, first choose the six who will be on the council, and then select people with "solo" designations, the remaining two will just be members
$\binom{10}{6}\cdot6*5*4*3 = 75600$