A certain panel is to be composed of...

[RonPurewal]

A student sent me this question, which I'm posting here for maximum benefit to the community.

#3 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

MGMAT explanation:

In the present case, we know that choosing three women from x + 2 women would yield 56 groups of 3. These numbers must correspond to a specific value of x. Do not worry if you do not know what value of x would yield these results (in this case, x must equal 6, because the only way to obtain 56 groups if choosing 3 is to choose from a group of 8. Since the statement tells us that 8 is 2 more than the value of x, x must be 6). The GMAT does not expect you to memorize all possible results. It is enough to understand the underlying concept: if you know the number of groups yielded (in this case 56), then you know that there is only one possible value of x.

The explanation provided doesn't make much sense to me. Actually, I don't understand the way it is figured out the value of x from st.1.

[RonPurewal]

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Here's an analogy to what they're doing: Let's say there's a 8.75% sales tax on the price of an item, and the question is "What's the price of the item?"

Statement 1 is like saying this:
(1) If the price of the item were $14.42 higher, the amount of sales tax charged on the item would be $22.35.
This would be a horrible, horrible thing to calculate, but you don't have to calculate it. The point is that there's going to be a single price for which the sales tax is $22.35 (since the tax keeps increasing as the prices increase). So, that's a single number"”we don't really care what number"”and so the desired value is whatever is $14.42 less than that number.

The same thing is true here. Just as there's exactly one price for which the tax is $22.35, there's going to be exactly one number of women that would give 56 combinations of three. So, statement 1 fixes the number of women"”it's just two less than whatever that is. No need to actually find it.

[RonPurewal]

The student wrote:

My naif way to figure x out from st.1 would be:

3w+2=56--->w=18

I'm aware, though, that I'm not taking in account the combinations/permutations situation + the other group involved in the permutation/combination. Indeed, I'm missing how to integrate and translate those info in the equation above.

Before you consider any of the mathematics below, remember that there's no need to calculate anything for this statement.
If you were actually required to calculate something, the problem would almost certainly be less laborious than this one.

So"”just to be redundant"”it is extremely unlikely that you'll ever have to do anything like the below. But, for the sake of completeness, here it is anyway.

* 56 is a small number, so, if I had to calculate this, I wouldn't bother with algebra. I'd just try modest-sized numbers and see what happens.
If there are six women, the number of combinations is (6*5*4)/3! = 20. Too small.
If there are seven women, the number of combinations is (7*6*5)/3! = 35. Still too small.
If there are eight women, the number of combinations is (8*7*6)/3! = 56. Boom. So you have 8 - 2 = 6 women.

[RonPurewal]

* If you set this up with algebra, it's an absolute mess. But, you asked for it, so here's how it works:
Say you have n women. In that case, the problem says that, if you had (n + 2) women, there would be 56 combinations of three.
The number of combinations of 3 women from a pool of (n + 2) is (n + 2)(n + 1)(n)/3!. So,
(n + 2)(n + 1)(n)/3! = 56
This is a third-degree equation, so there's no reasonable way to solve it other than to just plug in numbers. If you notice that 56 is 8 * 7, and there's 3! = 6 in the denominator, you may be able to see that n = 6 will work (it will give 8*7*6/3! = 8*7 = 56). So, n = 6.

I know this is the fourth time I'm saying the same thing, but, you don't have to be able to do this stuff. If a gmat problem involves calculations like these, then you probably don't have to do them in the first place.

[OliviaS193]

I just came across this question on the exam and the answer explanation stated this:

(1) INSUFFICIENT: This statement tells us that choosing 3 from x + 2 would yield 56 groups.

One concept that you need to know for the exam is that when dealing with combinations and permutations, each result corresponds to a unique set of circumstances. For example, if you have z people and know that choosing two of them would result in 15 different possible groups of two, it must be true that z = 6. No other value of z would yield exactly 15 different groups of two. So if you know how many subgroups of a certain size you can choose from an unknown original larger group, you can deduce the size of the larger group.

In the present case, we know that choosing three women from x + 2 women would yield 56 groups of 3. These numbers must correspond to a specific value of x. Do not worry if you do not know what value of x would yield these results (in this case, x must equal 6, because the only way to obtain 56 groups if choosing 3 is to choose from a group of 8. Since the statement tells us that 8 is 2 more than the value of x, x must be 6). The GMAT does not expect you to memorize all possible results. It is enough to understand the underlying concept: if you know the number of groups yielded (in this case 56), then you know that there is only one possible value of x.

I don't understand why it's insufficient? All explanations seem to say that it is actually sufficient information. Thanks!

[Sage Pearce-Higgins]

Statement (1) is sufficient to tell us the number of women, i.e. x. However, we still don't know the number of men, y. If y = 2, then there would be just one possible selection of men, but if y = 3, then there would be more possible selections of men. Therefore statement (1) alone is insufficient.

I agree that the explanation isn't quite clear here - it focuses only on the number of women, and doesn't mention the number of men.