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In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then $$ \Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...

I'm looking for a geometric or combinatorial depiction of the algebraic identity $$ xyz = \frac{1}{24} \Big\{(x+y+z)^3 - (x-y+z)^3 - (x+y-z)^3 + (x-y-z)^3 \Big\}. \label{*}\tag{$*$} $$ Here is the ...

This is a practical and very soft question, with the combinatorial database http://www.findstat.org in mind. I have a few, around 20, families of combinatorial objects, for example Dyck paths, ...

Theorem $[n]=\{1,\ldots,n\}$. Let $\lbrace (R_i, S_i), i \in I \rbrace, R_i, S_i \subset [n]$ be such that $R_i \cap S_i = \emptyset, R_i \cap S_j \ne \emptyset (i \ne j)$. Then $$\sum_{i \in I} \frac{...