Difficult P & C Problem

We should use the formula for choosing 2 out of x people, when order is also important --> x!/(x-2)! = 90 --> x=10

sukhman wrote:A student committee on academic integrity has 90 ways to select a president and vice-president from a groupof candidates. The same person cannot be both president and vice-president. How many students are in the group?

5
8
9
10
15

I've added answer choices, which the GMAT would provide.
We can plug in the answers for the total number of students.

Answer choice C: 9
Number of options for president = 9. (Any of the 9 students.)
Number of options for vice-president = 8. (Any of the 8 remaining students.)
To combine these options, we multiply:
9*8 = 72.
Too small.
Eliminate A, B and C.

Answer choice D: 10
Number of options for president = 10. (Any of the 10 students.)
Number of options for vice-president = 9. (Any of the 9 remaining students.)
To combine these options, we multiply:
10*9 = 90.
Success!

The correct answer is D.

Don't get carried away in the thought,thinking if it is permutation or combination problem.
lets keep fact straight.

There are X people initially;from these x,I have to choose 1 person,i can choose that person in xC1 ways.
(lets say,this choice was for president).Now since I have done that.I have to choose the VP.But since I cannot choose the same person again;I am left with x-1 people.So I have to choose 1 person from x-1 people.This can be done in (x-1)C1 ways.
Since both the things are required,it will be "and" or "X" relation

So
XC1 * (X-1)C1 = x(x-1)=90 //this is what the question says

x^2 -x-90 = 0
x^2-10x+9x-90=0
x(x-10)+9(x-10)=0
x=10
Ans



Note:There is nothing about order or permutation here.Its just that you cant choose the same person again.

sandeepraghuvanshi wrote:Don't get carried away in the thought,thinking if it is permutation or combination problem.
lets keep fact straight.

There are X people initially;from these x,I have to choose 1 person,i can choose that person in xC1 ways.

The spirit there is right, but if you don't want to think whether it's a perm or a comb, you probably wouldn't want to go straight to using the combinations formula, especially when (x choose 1) is just the same as x anyway.

If we're avoiding perms and combs, I'd say that

(# of people) * (# of people - 1) = 90

and since the # must be a positive integer, it must be 10.