Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 4 / 6E / iii / Solution

Write both nonnegative integers $n,k$ in base-p expansions, padding with zero digits to a common length $L$. By the preceding polynomial identity,Expanding the last product with the binomial theorem, each term is obtained by choosing an integer $j_i$ with $0\le j_i\le n_i<p$. Its exponent is ${\sum }_i{j}_i{p}^i$ and its coefficient is ${\prod }_i\left(\genfrac{}{}{0ex}{}{n_i}{j_i}\right)$. Uniqueness of base-p expansions says that the only way this exponent can equal $k$ is to choose $j_i=k_i$ for every $i$.

If some $k_i>n_i$, no such term exists and the coefficient is zero. Otherwise its coefficient is precisely the product below. Comparing the coefficient of $x^k$ proves Lucas theorem:with the convention $\left(\genfrac{}{}{0ex}{}{a}{b}\right)=0$ for $b>a$. This also covers $k>n$, and padding either expansion by further zero digits multiplies the product only by $\left(\genfrac{}{}{0ex}{}{0}{0}\right)=1$.