Question

Binomial Theorem, proof

Answer 1

It follows from the definition of the binomial coefficient:

$\dbinom nr=\dfrac{n!}{r!(n-r)!}$

So we have

$(n+1)\dbinom nr=(n+1)\dfrac{n!}{r!(n-r)!}=(r+1)\dfrac{(n+1)!}{(r+1)!(n-r)!}$

That is, $(n+1)$ gets absorbed into the numerator's factorial, and we introduct $(r+1)$ into the denominator. Now, $(n+1)-(r+1)=n-r$, so we get

$(n+1)\dbinom nr=(r+1)\dfrac{(n+1)!}{(r+1)!((n+1)-(r+1))!}=(r+1)\dbinom{n+1}{r+1}$

as required.