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

## Suggestion

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

What does the binomial theorem give as the value of $(a+b)^n$? What about $(1+a)^n$?

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

Could we split our group of people into two groups of size $n$?

If we take 1 person from the first group, how many people do you have to take from the second to choose $n$ people overall?

What if we take $r$ people from the first group?

1. Prove that $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$

Could we write down the two factorial representations of $\dbinom{n-1}{r-1}$ and $\dbinom{n-1}{r}$?

Or could we imagine a bag of $n$ objects, and picking $r$ from it?

$[...]$ 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 $1 \leq r \leq n-1$.

We’ve just found an expression for $\dbinom{n}{r}$ — how is this useful?