Question

there are 17 people in a club . How many ways can 3 of them be chosen as president,vice president 'and Secretary? how many can 2 of them be chosen to be on the student Senate ?

Answer 1

We have to choose 3 people at random and give them separate posts.

Just to choose 3 from 17 : use combination

To choose 3 and give separate posts: use permutation

So we have to use permutation.
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It can be done is 17P3 ways.
${}^{17} P_3 = \frac{17!}{(17-3)!} \\ = \frac{17!}{14!} \\ =17 \times 16 \times 15 \\ = 4080$

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From the chosen 3, you have to select 2 at random for Senate.

Choose 2 from 3.

Number of ways = 3C2 = 3! / [ 2! × (3-2)! ] = 3! / 2! = 3

You can do it in 3 ways.