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.)
_
Plug in some numbers:
ReplyDeleteLet 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...?
I do have to wonder whether I'm reading the notations right. For Quantity A, for example, the denominator is said to be
ReplyDelete5(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.