2017 AMC 8 Problem 5

Attempt Problem 5 of the 2017 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

5.

What is the value of the expression 1⋅2⋅3⋅4⋅5⋅6⋅7⋅81+2+3+4+5+6+7+8?\dfrac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}?
1020 1020
1120 1120
1220 1220
2240 2240
3360 3360
Answer: B
Concepts:factorial
Small hint:
First add the denominator.
Big hint:
After the denominator is known, cancel factors from the numerator instead of multiplying everything out.
Video solution:
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Written solution:
The denominator is 1+2+⋯+8=36=2⋅3⋅6.1+2+\cdots+8=36=2\cdot3\cdot6. Canceling those three factors from the numerator leaves 4⋅5⋅7⋅8=1120.4\cdot5\cdot7\cdot8=1120.
Thus, B is the correct answer.
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