2017 AMC 8 Problems/Problem 5

Problem 5

What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$ ?

$\textbf{(A) }1020\qquad\textbf{(B) }1120\qquad\textbf{(C) }1220\qquad\textbf{(D) }2240\qquad\textbf{(E) }3360$

Solution 1

We evaluate both the top and bottom: $\frac{40320}{36}$ . This simplifies to $\boxed{\textbf{(B)}\ 1120}$ .

Solution 2

It is well known that $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$ . Therefore, the denominator is equal to $\frac{8 \cdot 9}{2} = 4 \cdot 9 = 2 \cdot 3 \cdot 6$ . Now we can cancel the factors of $2$ , $3$ , and $6$ from both the numerator and denominator, only leaving $8 \cdot 7 \cdot 5 \cdot 4 \cdot 1$ . This evaluates to $\boxed{\textbf{(B)}\ 1120}$ .

Solution 3

We have 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40320, and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36/ And 40320/36 = 1120. So the answer is (B)1120

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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All AJHSME/AMC 8 Problems and Solutions

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2017 AMC 8 Problems/Problem 5

Contents

Problem 5

Solution 1

Solution 2

Solution 3

See Also