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

I'm struggling with two responses in the answers provided by MGMAT...

-- The number of panels will be equal to the number of groups of three that could be chosen from x women multiplied by the number of groups of two that could be chosen from y men.

-- 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.

Can you please explain why both these statements have to be true?

thanks much,
k

sure.

one:


this is just one example of something called the "fundamental counting principle", which states that when you make more than one independent decision, you simply multiply together the #'s of ways in which you can make each decision.
for instance, if you have 3 shirts and 4 pairs of pants, and you have to make an outfit at random (no style considerations) from one shirt and one pair of pants, then this can be done in 3 x 4 = 12 different ways.

here, you have 2 independent decisions: choosing the women and choosing the men. therefore, you just find the # of ways in which you could do each of these, and multiply those numbers together.

--

two:


just think about this, which should be clear after a little reflection:

the more you're choosing from, the more ways there will be to make your choice.

for instance: as noted above, there are 15 ways to choose two items/people from a set of six. therefore, there must be more than 15 ways to choose two items from a set of more than six, and fewer than 15 ways to choose two items from a set of fewer than six.

because the # of ways of choosing will increase monotonically with the # of available items, you have the result proffered in the explanation.

sweetness

Hi
Can you please post the answer as well to the same , I am getting to a value where x is not coming as a integer

How can any of the choices be sufficient.

Rahul

"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 "

- - - - - - - - - - - - - - - - - -

After thinking about this problem for a while, here is a way I found to approach it:

First we have to rephrase the question being asked. The number of possible 3-women, 2-men panels will be: (the number of possible groups of exactly three women) X (the number of possible groups of exactly two men). We see that to find both the number of possible 3-women groups and the number of possible 2-men groups, we first need to know the total number of women (given in the problem as "x") AND the total number men ("y"). We need to know BOTH totals to answer the question: without knowing x AND y, we don't know which two numbers to multiply to find the total number of possible 3-women, 2-men panels. This problem could therefore be rephrased as, "What are x AND y?" or "What are the total number of women AND the total number of men?"

Statement 1 (A): Without even doing any math, we can see that the statement only discusses women. Since no information is given regarding the total number of men (y), we can already determine this statement to be INSUFFICIENT. It doesn't tell us what both x AND y are. On our AD/BCE grid, we cross out A and D.

Statement 2 (B): This statement alone only gives us a relationship (ratio) between the total number of women and the total number of men. As the MGMAT Word Translations book says, ratios "do not provide enough information, on their own, to determine the exact quantity for each item." Since the problem itself doesn't give us a value for either x or y (which is why we're looking for them), we can't solve for either variable and therefore this statement alone is INSUFFICIENT. On our AD/BCE grid, we cross out B.

We are now left with possible answer choices of either (C) or (E).

Statements 1 & 2 Together (C): Statement 1 says that if two more women were added to the current total of women (x), then 56 different groups of exactly three women would be possible. If we know the possible number of groups (56) and the number of women we were picking (3) from the total number of women PLUS TWO more women (x+2), it is possible now to work backwards to find x. Since we can find x, we can then apply the formula given in Statement 2 to find out y. Since Statements 1 and 2 together allow us to find out both x AND y, we can answer our rephrased question (without actually having to solve for them of course) and therefore the answer is (C): both statements together are SUFFICIENT.

This works for me, but I'm open to anyone finding some holes in the reasoning above or suggestions to strengthen it.

Could you explain how you did the backsolving to come up with x? I know that you don't need to solve down that far to realize the answer is C, but I the backsolving aspect will be helpful to know.
Thanks!

The principle you quoted in your original post is the key: "...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."

Since Statement (1) tells us that 56 different (sub)groups of three women (a certain size) can be chosen from x+2 women (an unknown original larger group), we can deduce the size of the larger group.

The answer is that x + 2 = 8, because 56 different combinations of three women is only possible if the larger group is 8:

8! 8 x 7 x 6 x 5 x 4 x 3 x 2 8 x 7 x 6
---------- = ------------------------------ = ------------- = 8 x 7 = 56
(3!) (5!) (3 x 2) x (5 x 4 x 3 x 2) 6


If x + 2 = 8, that means x (the total number of women) = 6. Since the total number of women is one more than the number of men (x=y+1), the total number of men is 5.

Thank you! I appreciate it!

nice work. more calculation than you need for a data sufficiency question (you don't need the actual number and can trust that it will come out to be an integer if it is about people). but it's always good to grasp the concepts.