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Difficult P & C Problem

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by sukhman » Thu Sep 12, 2013 8:07 am
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? Answer 10
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by Java_85 » Thu Sep 12, 2013 9:21 am
We should use the formula for choosing 2 out of x people, when order is also important --> x!/(x-2)! = 90 --> x=10
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by Brent@GMATPrepNow » Thu Sep 12, 2013 9:46 am
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? Answer 10
Slightly different approach, same result.

Let's let x = the number of students.

Take the task and break it into stages.

Stage 1: Select a president
There are x students to choose from, so we can complete stage 1 in x ways

Stage 2: Select a vice-president
There are x-1 students remaining, so we can complete stage 2 in x-1 ways

By the Fundamental Counting Principle (FCP) we can complete both stages (x)(x-1) ways

Since we are told that there are 90 ways to select a president and vice-president, we can conclude that: (x)(x-1) = 90

At this point, there would be answer choices, so we could just start plugging in values for x, to get x = 10

Cheers,
Brent

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775
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by GMATGuruNY » Thu Sep 12, 2013 11:59 am
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.
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by sandeepraghuvanshi » Thu Dec 01, 2016 11:39 am
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
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Note:There is nothing about order or permutation here.Its just that you cant choose the same person again.
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by Matt@VeritasPrep » Thu Dec 08, 2016 8:39 pm
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.
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