BTGmoderatorDC wrote:Which of the following is an integer?
I. 12! / 6!
II. 12! / 8!
III. 12! / 7!5!
A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III
OA E
Source: GMAT Prep
Before actually solving this problem, let's review how factorials can be expanded and expressed. As as example, we can use 5!.
5! could be expressed as:
5!
5 x 4!
5 x 4 x 3!
5 x 4 x 3 x 2!
5 x 4 x 3 x 2 x 1!
Understanding how this factorial expansion works will help us work our way through each answer choice, especially answer choices 1 and 2.
I. 12!/6!
Since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6!
Plugging this in for answer choice 1, we have:
(12 x 11 x 10 x 9 x 8 x 7 x 6!)/6! = 12 x 11 x 10 x 9 x 8 x 7, which is an integer.
II. 12!/8!
Once again, since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8!
Plugging this in for answer choice 2, we have:
(12 x 11 x 10 x 9 x 8!)/8! = 12 x 11 x 10 x 9, which is an integer.
III. 12!/(7!5!)
Once again, since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8 x 7!
Plugging this in for answer choice 3 gives us:
(12 x 11 x 10 x 9 x 8 x 7!)/(7!5!)
(12 x 11 x 10 x 9 x 8)/(5 x 4 x 3 x 2 x 1)
We strategically combine the numbers in the denominator so that cancellation with those in the numerator will be easy:
(12 x 11 x 10 x 9 x 8)/(12 x 10 x 1)
11 x 9 x 8, which is an integer.
We see that the quantities in Roman numerals I, II and III are all integers.
Alternate solution:
For any positive integers m, n and p,
1) If m > n, then m!/n! is always an integer (which is, in fact, mP(m - n)).
2) If m = n + p, then m!/(n!p!) is always an integer (which is in fact mCn or mCp).
From the above two facts, we see that all three quotients in the Roman numerals must be integers.
Answer: E
