Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 1 / e / iv

Choose the image of a function as a $j$-element subset of $B$, then choose a surjection onto that selected image as in (iii). ThereforeThe answer is zero when $j>n$. If $j=0$, it is one for $m=0$ and zero for $m>0$, as the Stirling numbers of the second kind conventions require.