ratio DS MGMAT

[rommysingh]

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

[GMATGuruNY]

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

Statement 1:
No information about the number of men.
INSUFFICIENT.

Statement 2:
Since x and y can be any two consecutive nonnegative integers, INSUFFICIENT.

Statements combined:
Statement 1:
Since 56 = 7*8, at least 7 women are required to yield 56 different groups of 3.
TEST CASES:
From 7 women, the number of groups of 3 that can be formed = 7C3 = (7*6*5)/(3*2*1) = 35.
From 8 women, the number of groups of 3 that can be formed = 8C3 = (8*7*6)/(3*2*1) = 56.
From 9 women, the number of groups of 3 that can be formed = 9C3 = (9*8*7)/(3*2*1) = 84.
As indicated by the red case, 8 women are required to form 56 different groups of 3.
Since 8 is equal to two more than the ACTUAL number of women, x = 8-2 = 6.

Statement 2:
Since x=6 and x = y+1, y=5.

Since the values of x and y are known, it is possible to determine the number of 3-women, 2-men panels that can be formed.
SUFFICIENT.

The correct answer is C.

[Brent@GMATPrepNow]

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

Target question: How many different panels can be formed with these constraints?

First recognize that, since the order of the selected people does not matter, we can use combinations to solve this. We can select 3 women from x women in xC3 ways, and we can select 2 men from y men in yC2 ways. So, the total number of possible panels = (xC3)(yC2)

As you can see,the answer to the target question will depend solely on the individual values of x and y.

Statement 1: If two more women were available for selection, exactly 56 different groups of three women could be selected.
Since there's no information about the number of men, statement 1 is NOT SUFFICIENT

Statement 2: x = y + 1
There are several pairs that meet this condition. Here are two:
Case a: x = 3 and y = 2, in which case there's only 1 possible panel (since we'd have no choice but to select all 5 people)
Case b: x = 201 and y = 200, in which case there are TONS of possible panels
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 basically tells us that (x+2)C3 = 56 ["x+2 CHOOSE 3 equals 56"]
Do we need to solve for x? NO. We need only recognize that we COULD solve for x.
Let's start checking a few possible values.
3C3 = 1
4C3 = 4
5C3 = 10
6C3 = 20

Aside: if anyone is interested, we have a free video on calculating combinations (like 6C3) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

We can see that the numbers keep increasing, so there will be ONLY ONE value of x such that (x+2)C3 = 56 [incidentally, 8C3 = 56. So, (x+2) = 8, which means x = 6. Of course, that doesn't really matter since we need only recognize that we COULD determine the value of x]
So, from statement 1, we COULD determine the value of x
Once we know the value of x, we can use statement 2 to determine the value of y.
At this point, we can answer the target question with certainty, so the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

[nikhilgmat31]

Using Statement 1
if two more women are availbe & number of ways of 3 groups are 56

x+2C3 = 56

(x+2)! / (3! * (x+2-3)! = 56

(x+2)(x+1)(x) (x-1)! / ( x-1)! = 56 * 6

(x+2)(x+1)(x) = 56 * 6
(x+2)(x+1)(x) = 6* 7 *8

x = 6

But Statement 1 gives only value of X i.e. number available women so Non Sufficient

Statement 2
x = y + 1

so y =5

now we can find 6C3 * 5C 2 Answer is C - Both statements together are sufficient.