My first problem

Relevant wiki: Factorials

let $107=N$

then the product stated is

$=\frac{N!^{N}}{1!2!3!...N!}=\frac{1^{N}2^{N}3^{N}...N^{N}}{1!2!3!...N!}$

Now we reach the vital step, look at the denominator, N factorials have a common factor of 1, N-1 factorials have a common factor of 2, N-2 factorials have a common factor of 3 and so on. if you group the repeated numbers in the denominator you can see that we can write the above expression as

$\frac{1^{N}2^{N}3^{N}...N^{N}}{1^{N}2^{N-1}3^{N-2}...N^{1}}=\frac{1^{N}}{1^{N}}\cdot\frac{2^{N}}{2^{N-1}}\cdot\frac{3^{N}}{3^{N-2}}...\frac{N^{N}}{N^{1}}=1^{0}2^{1}3^{2}...N^{N-1}=\prod_{k=1}^{N} k^{k-1}$ .