2016 AMC 12A Problem 1

Attempt Problem 1 of the 2016 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12A solutions, or check the answer key.

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

1.

What is the value of 11!−10!9!?\dfrac{11!-10!}{9!}?
9999
100100
110110
121121
132132
Answer: B
Concepts:factorialfactoring
Small hint:
Factor 10!10! out of the numerator
Big hint:
11!−10!=10!(11−1)11!-10!=10!(11-1) and 10!=10⋅9!10!=10\cdot 9!
Solution:
Factoring the numerator gives 11!−10!9!=10!(11−1)9!=10⋅9!⋅109!=100. \begin{gathered} \dfrac{11!-10!}{9!}\\ =\dfrac{10!(11-1)}{9!}\\ =\dfrac{10\cdot 9!\cdot 10}{9!}\\ =100. \end{gathered}
Thus, the correct answer is B.
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