Which of the following is an integer?

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)

(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.

2) If m = n + p, then m!/(n!p!) is always an integer (which is in fact mCp).

From the above two facts, we see that all three quotients in the Roman numerals must be integers.

Answer: E