2020 AMC 8 Problems/Problem 12

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For positive integers $n$, the notation $n!$ denotes the product of the integers from $n$ to $1$. What value of $N$ satisfies the following equation? \[5!\cdot 9!=12\cdot N!\]

Solution 1

Notice that $5!$ = $2*3*4*5,$ and we can combine the numbers to create a larger factorial. To turn $9!$ into $10!,$ we need to multiply $9!$ by $2*5,$ which equals to $10!.$

Therefore, we have

\[10!*12=12*N!.\] We can cancel the $12$'s, since we are multiplying them on both sides of the equation.

We have

\[10!=N!.\] From here, it is obvious that $N=\boxed{10\textbf{(A)}}.$

-iiRishabii