Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 2 / e / i / Solution

For $n\ge 2$ and $1\le k\le n-1$, the factorial formula for a binomial coefficient givesThis proves Pascal's identity. The boundary cases $k=0,n$ follow for $n\ge 1$ using the usual extension $\left(\genfrac{}{}{0ex}{}{m}{j}\right)=0$ for $j<0$ or $j>m$. The recurrence itself requires $n\ge 1$.