## Tuesday, October 18, 2011

### this week's MGRE Math Beast Challenge problem

This week's problem is titled "Yippee!" I'm guessing the important element is the exclamation point, which indicates that we're dealing with factorials. Remember those?

5! (pronounced "five factorial") = 5 x 4 x 3 x 2 x 1 = 120 (Whoa-- I see that Wikipedia picked the same example.)

If you have a fraction like 6!/5!, you automatically know you can cancel out everything up to 5!. Behold:

(6)(5)(4)(3)(2)(1)/(5)(4)(3)(2)(1) = 6.

As a rule, then, x!/(x-1)! = x. This might be good to remember, along with the fact that you can't do factorials with negative integers.

Onward!

The MGRE problem this week:

"Yippee!"

y ≥ 2

Quantity A          Quantity B

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Go to it. My answer will appear in the comments. Oh, yes: keep in mind that zero factorial, or 0!, is defined as equal to 1. Note, too, that 1! = 1 as well. Why? Because factorials are supposed to help a person figure out permutations and combinations. If there are zero items in a set (null set), then there's only one possible "arrangement" of zero, hence 0! =1. Same with the number 1: if all you've got is a 1 in your set, then once again there's only one possible "arrangement." That's why 1! also equals 1.

(And in case you're wondering, yeah, I did have to look all that up.)

_

Kevin Kim said...

Plug in some numbers:

Let y = 3

A: 3!/(5)(3-2)! = (3*2*1)/(5*1) = 6/5

B: (3+1)/(3)(3-1) = 4/6 = 2/3

A is greater than B.

Let y = 4

A: 4!/5(2!) = (4*3*2*1)/(5*2!) = 24/10 = 12/5

B: (4+1)/(3)(3!) = 5/18

A is greater than B.

Consistent thus far. Let's go back to 2, then.

Let y = 2.

A: 2!/(5)(0!) = 2/(5)(1) = 2/5

B: (2+1)/(3)(2-1)! = 3/3(1!) = 1

B is greater than A.

Hmmm. Looks as if the answer is D yet again. Did I miss something?

Do keep in mind that, in quantity B, the numerator of the fraction is NOT a factorial. That's easy to miss.

QED...?

Kevin Kim said...

I do have to wonder whether I'm reading the notations right. For Quantity A, for example, the denominator is said to be

5(y - 2)!

In my reading of the problem, the factorial symbol applies only to the parenthetical quantity. Am I right to read it this way? What if I'm actually supposed to read that as

(5(y - 2))!

?

It makes a difference. If y = 4, then, by my initial reading, the numerator of Quantity A should be

(5)*((4-2)!), or 5*2!, or 10. But if we read the expression differently, we get

(5(4 - 2))!, which is 10!, and that's a damn huge number.

Conclusion: I think my initial reading is correct, because ETS/MGRE wouldn't fuck with us so cruelly.