Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 2 / a

There are $n(n-1)\cdots (n-k+1)$ ordered selections without replacement. Each unordered $k$-element subset occurs in exactly $k!$ orders, so the binomial coefficient isOne precise specification of its domain, codomain, and map isHere ${\mathbb{N}}_0=\{0,1,2,\dots \}$, and $0!=1$ includes the single empty selection. The actual values on $D$ are positive, so ${\mathbb{N}}_{+}$ is also a valid codomain. For fixed $n$, one can instead use the function $C_n:\{0,\dots ,n\}\to {\mathbb{N}}_{+}$ with $C_n(k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)$.

Apply the factorial formula for the binomial coefficients:For $n=0$ or $n=1$, the same equality holds with the conventional zero value when the lower index exceeds the upper index.