# In how many ways can the officers of the student council be selected?

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?

1. $$151200$$
2. $$75600$$
3. $$37800$$
4. $$18900$$

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?

$$\binom{10}{6}\cdot6*5*4*3 = 75600$$