GMAT Practice Question: Which of the following is an integer? I. \frac{12!}{6!} II. \frac{12!}{8!} III....
Question
Which of the following is an integer? I. \frac{12!}{6!} II. \frac{12!}{8!} III. \frac{12!}{7!5!}
- I only
- II only
- III only
- I and II only
- I, II, and III
Topics: factorials, fractions, calculations
Solution
Step 1
Statement I: We simplify \frac{12!}{6!} by canceling the common factorial.
\frac{12!}{6!} = \frac{12\times11\times10\times9\times8\times7\times\cancel{6!}}{\cancel{6!}} = 12\times11\times10\times9\times8\times7
Step 2
Since the result of Statement I is a product of integers, it is an integer.
12\times11\times10\times9\times8\times7\text{ is an integer}
Step 3
Statement II: We simplify \frac{12!}{8!} by canceling the common factorial.
\frac{12!}{8!} = \frac{12\times11\times10\times9\times\cancel{8!}}{\cancel{8!}} = 12\times11\times10\times9
Step 4
Since the result of Statement II is a product of integers, it is an integer.
12\times11\times10\times9\text{ is an integer}
Step 5
Statement III: We simplify \frac{12!}{7!5!} by canceling 7! and then checking divisibility by 5!.
\frac{12!}{7!5!} = \frac{12\times11\times10\times9\times8\times\cancel{7!}}{\cancel{7!}\times5!} = \frac{12\times11\times10\times9\times8}{5!}
Step 6
We factor numerator and denominator to cancel common factors
\frac{12\times11\times10\times9\times8}{5!} = \frac{12\times11\times10\times9\times8}{5\times4\times3\times2\times1}. Cancel 12/4=3, 10/5=2, 9/3=3, 8/2=4, so we get 3\times11\times2\times3\times4\text{ which is an integer}
Answer
E. I, II, and III are all integers.
