Binomial coefficient congruence across a prime row
= Binomial coefficient congruence across a prime row
{title2=$\binom{p+k}{r}\equiv\binom kr\pmod p\quad(0\leq r\leq k<p)$}
In polynomial <modular arithmetic>, <prime-row binomial coefficient divisibility> gives $(1+X)^p\equiv1+X^p$. Multiply by $(1+X)^k$ and compare coefficients of $X^r$ for $r<p$; the $X^p$ term cannot contribute.
// Target: foundations-of-mathematics.bigb