2020 AMC 8 Problem 12

Attempt Problem 12 of the 2020 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 2020 AMC 8 solutions, or check the answer key.

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

12.

For a positive integer n,n, the factorial notation n!n! represents the product of the integers from nn to 1.1. For example:
6!=6×5×4×3×2×1 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1
What value of NN satisfies the following equation?
5!×9!=12×N! 5! \times 9! = 12 \times N!
1010
1111
1212
1313
1414
Answer: A
Concepts:factorial
Small hint:
Cancel the factor 1212 from 5!5!
Big hint:
Rewrite 5!5! so the factor 1212 cancels cleanly
Video solution:
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Written solution:
Because 5!=120=12⋅10,5!=120=12\cdot10, the equation becomes 12⋅10⋅9!=12⋅N!. 12\cdot10\cdot9!=12\cdot N!. Canceling 1212 gives N!=10⋅9!=10!,N!=10\cdot9!=10!, so N=10.N=10.
Thus, the correct answer is A.
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