Problem:
For a positive integer n, the factorial notation n! represents the product of the integers from n to 1. For example:
6!=6×5×4×3×2×1
What value of N satisfies the following equation?
5!×9!=12×N!
Answer Choices:
A. 10
B. 11
C. 12
D. 13
E. 14
Solution:
Note first that
n!=n⋅(n−1)⋅(n−2)⋯1=n((n−1)⋅(n−2)⋯1)=n⋅(n−1)!.
With that in mind, further observe that
5!⋅9!=5⋅4⋅3⋅2⋅1⋅9!=120⋅9!=12⋅(10⋅9!).
Since 12⋅N!=12⋅(10⋅9!), we know N!=10⋅9!.
Using our note from above, we know that 10⋅9!=10!, so N=10.
Thus, the correct answer is A.
Answer: A.
The problems on this page are the property of the MAA's American Mathematics Competitions