Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Discrete mathematics / 2 / b / Solution

Use mathematical induction on $n\ge 0$. For $n=0$, the sum is $0!0=0=1!-1$. Suppose it is $(n+1)!-1$ at $n$. ThenThe induction is complete, proving ${\sum }_{i=0}^{n}i!i=(n+1)!-1$. The factorial identity $i!i=(i+1)!-i!$ also explains the cancellation directly.