Use Mathematical Induction to prove
\dbinom{n}{2}a^{n-2}b^2+\dots+\dbinom{n}{r}a^{n-r}b^r+\dots+ \dbinom{n}{n-1}ab^{n-1} + b^n\begin{align*}
&(a+b)^{k+1}\
&= (a+b)(a+b)^k\
&= (a+b)\Bigg(a^k + \dbinom{k}{1}a^{k-1}b + \dbinom{k}{2}a^{k-2}b^2+\dots
+\dbinom{k}{r}a^{k-r}b^r +\dots+ \dbinom{k}{k-1}ab^{k-1} + b^k \Bigg)\
&\begin{aligned}[t]
&= a^{k+1} + \Bigg[1+\dbinom{k}{1}\Bigg]a^kb + \Bigg[\dbinom{k}{1}+\dbinom{k}{2}\Bigg]a^{k-1}b^2 + \dotsb\
&\dotsb + \Bigg[\dbinom{k}{r-1} + \dbinom{k}{r}\Bigg]a^{k-r + 1}b^r + \dotsb + \Bigg[\dbinom{k}{k-1} + 1\Bigg]ab^{k} + b^{k+1}.
\end{aligned}
\end{align*}$$
3. Use
4. QED.