Question

How many different ways can the students at a school select the president, vice president and secretary from a group of 5?

Answer 1

Answer: 60 ways.
Explanation: Suppose that we had 5 people, A, B, C, D, and E, and we had to select a P, VP, and S.

For the President, we have 5 different selections.
If we choose a president, then we can only pick 1 from 4 people for the VP.
Using the same logic, we'd have 3 different choices for the S.

Hence, we have 5 · 4 · 3 different ways.

Alternatively, this is:

$\frac{5!}{(5 - 3)!} = ^{5}P_3$ different ways.

Answer 2

 

P(5,3) .... or 5P3 if your textbook likes it that way ... = (5)(4)(3) = 60 ways.  

If you don't know about the permutation counting function, P(n,k) is the number of ways to pick k out of n different items, where order matters.  

P(n,k) = n(n-1)(n-2)...(n+1-k)  

It's easier to write with factorials, with P(n,k) = n!/k!, but the above is easier to compute most of the time.  

Without that, you can still solve with common sense. There are 5 ways to pick the president. For each of those, there are 4 choices left for vice president, and for each of those there are 3 choices left for secretary. 5*4*3 =

60 total