Lucas Correspondence Theorem

Let $p$ be Prime and

$$r = r_mp^m+\ldots+r_1p+r_0 \qquad (0\leq r_i<p)$$ (1)

$$k = k_mp^m+\ldots+k_1p+k_0 \qquad (0\leq k_i<p),$$ (2)

then

$${r\choose k}=\prod_{i=0}^m {r_i\choose k_i} {\rm\ (mod\ } p).$$ (3)

This is proved in Fine (1947).

References

Fine, N. J. ``Binomial Coefficients Modulo a Prime.'' Amer. Math. Monthly 54, 589-592, 1947.