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