Review question

# Can we show $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-2}{r-1}+\binom{n-3}{r-1}+\dotsb+\binom{r-1}{r-1}$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8014

## Question

1. Prove that $\sum_{r=0}^{n}\binom{n}{r}=2^n.$

2. By considering the number of ways of choosing a set of $n$ people from a set of $2n$ people, or otherwise, prove that $\binom{2n}{n}=\binom{n}{0}^2+\binom{n}{1}^2+\dotsb+\binom{n}{n}^2.$

3. Prove that $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$ and hence, or otherwise, prove that $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-2}{r-1}+\binom{n-3}{r-1}+\dotsb+\binom{r-1}{r-1},$ where $r$, $n$ are positive integers with $1 \leq r \leq n-1$.