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

I. 12!/6! = (12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)/(6)(5)(4)(3)(2)(1) = (12)(11)(10)(9)(8)(7) = some integer

II. 12!/8! = (12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)/(8)(7)(6)(5)(4)(3)(2)(1) = (12)(11)(10)(9) = some integer

III. 12!/7!5! = (12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)/(7)(6)(5)(4)(3)(2)(1)(5)(4)(3)(2)(1) = (12)(11)(10)(9)(8)/(5)(4)(3)(2)(1)
= (12)(11)(2)(9)(8)/(4)(3)(2)(1)
= (11)(2)(9)(8)/(2)(1)
= (11)(9)(8)
= some integer

Answer: E

Cheers,
Brent

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