This concept was presented to me: (2k+3)!/(k+2)! is the product of integers from k+3....(2k+3).
I do not understanding why this is. I tried using the definition of factorial on numerator and denominator to try and reach something that looked like this but could not. Could someone please explicitly show me (and explain) why this equality holds? Thanks.
Generally, if $n>k$, we have that $\frac{n!}{k!}$ is the product of all the numbers from $k+1$ to $n$. Too see why, consider a small example, such as:
$$\frac{7!}{3!}=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1}$$
Note that the entire denominator cancels, leaving just the product of numbers from $4$ to $7$.
Does this help?