# A sum of binomial coefficients modulo p

## (The Proof of (11))

**(11)**

* Proof:*

By induction on *n*: For
we must have *n=k* and the only possible value of *m* in the sum is *j*,
so that the result is trivial. Now assume that , and write
*m* and *n* in base *p*. Then
**(25)**

for each *m* in the sum in (11), as for each *i*.
Thus, by Lucas' Theorem, the sum in (11) is congruent to

where the sum is over all -tuples of integers
satisfying (25) and not all zero. This
is exactly the sum of the coefficients of
in
,
which equals

(11) then follows from the induction hypothesis as
and .