This tag is for basic questions about the study of finite or countable discrete structures — specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions. ...

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Hypergraph rainbow colouring of $\{1 \dots n\}$ for $A = \{A_1, \dots A_k\} : A_i \subset \{1, \dots n\}$

We are given collection of sets $A = \{A_1, \dots A_k\}$, where each set $A_i \subset \{1,\dots n\}$. Colouring $\{1, \dots n\}$ into $s$ colours would be called 'rainbow' for given $A$, if $\forall ...
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An example where $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is the number of ways of counting something?

Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer. There is a answer given here to this question here. I've seen how it can be proven using recurrence ...