Which of the following is an integer? I. (12!)/(6!) II. (12!)/(8!) III. (12!)/(7!5!)

I, II, and III (E). Statement I: We simplify (12!)/(6!) by canceling the common factorial.

GMAT Number Properties: Which of the following…

Which of the following is an integer? I. \frac{12 !}{6 !} II. \frac{12 !}{8 !} III. \frac{12 !}{7 ! 5 !}

factorialsfractionscalculations

Explanation	Calculations	Help
$Statement I: We simplify \frac{12 !}{6 !} by canceling the common factorial.$	$\frac{12 !}{6 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 !} = 12 \times 11 \times 10 \times 9 \times 8 \times 7$
$Since the result of Statement I is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 \times 8 \times 7 is an integer$
$Statement II: We simplify \frac{12 !}{8 !} by canceling the common factorial.$	$\frac{12 !}{8 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 !}{8 !} = 12 \times 11 \times 10 \times 9$
$Since the result of Statement II is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 is an integer$
$Statement III: We simplify \frac{12 !}{7 ! 5 !} by canceling 7! and then checking divisibility by 5! .$	$\frac{12 !}{7 ! 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 ! \times 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 !}$
$We factor numerator and denominator to cancel common factors$	$\frac{12 \times 11 \times 10 \times 9 \times 8}{5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} . Cancel 12/4 = 3, 10/5 = 2, 9/3 = 3, 8/2 = 4, so we get 3 \times 11 \times 2 \times 3 \times 4 which is an integer$

I, II, and III (E)