Review question

# Can we prove these four binomial coefficient identities? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6102

## Question

By considering the expansion of $(1+x)^n$ where $n$ is a positive integer, or otherwise, show that:

1. $\dbinom{n}{0}+\dbinom{n}{1}+\dbinom{n}{2}+\dotsb+\dbinom{n}{n}=2^n$;

2. $\dbinom{n}{1}+2\dbinom{n}{2}+3\dbinom{n}{3}+\dotsb+n\dbinom{n}{n}=n2^{n-1}$;

3. $\dbinom{n}{0}+\dfrac{1}{2}\dbinom{n}{1}+\dfrac{1}{3}\dbinom{n}{2}+\dotsb+\dfrac{1}{n+1}\dbinom{n}{n}=\dfrac{1}{n+1}(2^{n+1}-1)$;

4. $\dbinom{n}{1}+2^2\dbinom{n}{2}+3^2\dbinom{n}{3}+\dotsb+n^2\dbinom{n}{n}=n(n+1)2^{n-2}$.