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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Difficult P & C Problem
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Slightly different approach, same result.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
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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I've added answer choices, which the GMAT would provide.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
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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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.
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.
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.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.
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.
