Let $n_1, n_2$ be two positive integers. It is well known that the binomial coefficient, $$ \frac{(n_1+n_2)!}{n_1! \, \, n_2!} $$ represents the number of possible choices of $n_1$ elements from a set containing $n_1+n_2$ elements.
Moreover, it is well known that, if $n$ is an even integer, then $(n-1)!!$ can be interpreted as the number ways the elements of the set $\{1, \ldots, n\}$ can be divided into $n/2$ pairs.
Do $$ \frac{(n_1+n_2-1)!!}{(n_1-1)!!(n_2-1)!!}, $$ or $$ \frac{(n_1+n_2-2)!!}{(n_1-1)!!(n_2-1)!!}, $$ which combines the two formulas above, have some combinatorial interpretation, similarly to the two quantities which have been introduced above?