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Which of the following is an integer? I. \frac{11 !}{4 !} II. \frac{11 !}{5 !} III. \frac{11 !}{6 ! \times 5 !}

Explanation	Calculations	Help
$Statement I: We simplify \frac{11 !}{4 !} by canceling the common factorial.$	$\frac{11 !}{4 !} = \frac{11 \times 10 \times 9 \times 8 \times 4 !}{4 !} = 11 \times 10 \times 9 \times 8$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement I is a product of integers, it is an integer.$	$11 \times 10 \times 9 \times 8 is an integer$
$Statement II: We simplify \frac{11 !}{5 !} by canceling the common factorial.$	$\frac{11 !}{5 !} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 5 !}{5 !} = 11 \times 10 \times 9 \times 8 \times 7$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement II is a product of integers, it is an integer.$	$11 \times 10 \times 9 \times 8 \times 7 is an integer$
$Statement III: We simplify by canceling and then simplifying the remaining fraction.$	$\frac{11 !}{6 ! \times 5 !} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 ! \times 5 !} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We calculate the numerator and divide by 5! to confirm it is an integer.$	$11 \times 10 = 110 110 \times 9 = 990 990 \times 8 = 7920 7920 \times 7 = 55440 5! = 120 \frac{55440}{120} = 462 is an integer$

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E

Which of the following is an integer? I. \frac{11 !}{4 !} II. \frac{11 !}{5 !} III. \frac{11 !}{6 ! \times 5 !}

Explanation	Calculations	Help
$Statement I: We simplify \frac{11 !}{4 !} by canceling the common factorial.$	$\frac{11 !}{4 !} = \frac{11 \times 10 \times 9 \times 8 \times 4 !}{4 !} = 11 \times 10 \times 9 \times 8$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement I is a product of integers, it is an integer.$	$11 \times 10 \times 9 \times 8 is an integer$
$Statement II: We simplify \frac{11 !}{5 !} by canceling the common factorial.$	$\frac{11 !}{5 !} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 5 !}{5 !} = 11 \times 10 \times 9 \times 8 \times 7$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement II is a product of integers, it is an integer.$	$11 \times 10 \times 9 \times 8 \times 7 is an integer$
$Statement III: We simplify by canceling and then simplifying the remaining fraction.$	$\frac{11 !}{6 ! \times 5 !} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 ! \times 5 !} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We calculate the numerator and divide by 5! to confirm it is an integer.$	$11 \times 10 = 110 110 \times 9 = 990 990 \times 8 = 7920 7920 \times 7 = 55440 5! = 120 \frac{55440}{120} = 462 is an integer$

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E