Question:
Let $G=(V,E)$ be a graph with $n$ connected components, in which each component $G_i$ is the $K_{2i}$ graph. How many perfect matches are there in $G$?
$Solution.$
We shall have a perfect match in each connected component. In addition, we know that the number of all perfect matches over $K_{2i}$ is $\prod _{k=0}^{i} 2i-2k+1$, because if we pick a vertex $v$, then for a match we have $2i-1$ vertices, then for another vertex $u$ we will have $2i-3$ options, and so on... We want simultaneously to have a perfect match in each component and therefore we have: $$\left(\prod _{k=0}^{1} 2-2k+1\right) \cdotp \left(\prod _{k=0}^{2} 4-2k+1\right) \cdots \left(\prod _{k=0}^{n} 2n-2k+1\right) =\prod _{i=0}^{n}\prod _{k=0}^{i}2i-2k+1$$ options.
Now, it seems that I am missing something here, or perhaps I didn't understand this question well. Therefore, I will be grateful for some help.
The product $\prod_{k=1}^i 2i - 2k + 1$ is often denoted $(2i-1)!!$; this is the product of odd numbers less than $2i$. With this notation, your expression becomes the more palatable $$\prod_{i=1}^n (2i-1)!!.$$ This can also be written as $$ \prod_{i=1}^n (2i-1)^{n-i+1},$$ which just says that in the above expression, the $i$-th odd number appears $n-i+1$ times.