Question : Selecting a Panel
Source : Manhattan GMAT CAT

A certain panel is to be composed of exactly three women and exactly two men, chosen from x women and y men. How many different panels can be formed with these constraints?

(1) If two more women were available for selection, exactly 56 different groups of three women could be selected.

(2) x = y + 1

My approach was through slot method :

___ ___ ___ and ___ ___
3 women out of x and 2 men out of y

x * (x-1) * (x-2) and y * (y-1)

so we got to find what does the above expression evaluate to ?

1st statement -> if two more women were available so x is now x+2

so women can be chosen in (x+2) * (x+1) * (x) ways
and men can be chosen in (y) * (y-1) ways

[(x+2) * (x+1) * (x)] * [((y) * (y-1)] = 56
however, this expression won't evaluate to a definite answer hence it is INSUFFICIENT which rules out options A and D

2nd statement -> x = y+1
Putting y+1 as the value of x in the above expression in blue again wont give me a definite value for the expression hence this statement is INSUFFICIENT which rules out option B

Combining both, we get two equations and we have two variables. Although I did not attempt to solve the equations but I think that it will give me the required value for x and y and hence we will be able to evaluate the expression in blue on top and hence combining statement 1 and statement 2 it is SUFFICIENT and hence answer is C.

Is my approach incorrect ?

Regards

well, you can do this problem with a lot less work than that.

1/
no information about the number of men ("y"), so there are an infinite number of possibilities.

2/
x and y could be, say, 5 and 4. or they could be 1,000,000 and 999,999. or any of infinitely many other possibilities. these are clearly going to give different final answers, so, insufficient.

together/
from statement 1 you can tell that you're going to get A SPECIFIC # of women.
i.e., whenever you add more women, there are more ways to choose 3 of them. so, if you have a specific # of groups possible (here 56), that fixes the number you're choosing from. since this is data sufficiency, you don't have to find that actual number.
then, from statement 2, you'll also get a specific # of men.
thus you'll have specific numbers all around, so you'll be able to get a specific value for the solution.
sufficient.

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as far as your slot method:




this setup is incorrect.
a "panel" is a situation in which order doesn't matter (since there is no particular order, and there are no distinct positions, on a "panel"). so, your product for the women should be divided by 3!, and your product for the men should be divided by 2!.

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