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!â‹…9!=5â‹…4â‹…3â‹…2â‹…1â‹…9!=120â‹…9!=12â‹…(10â‹…9!).
Since 12â‹…N!=12â‹…(10â‹…9!), we know N!=10â‹…9!.
Using our note from above, we know that 10â‹…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