Indian Statistical Institute (ISI) B.Math & B.Stat : Combinatorics

For $ k \geq 1$, find the value of \[ \binom{n}{0}+ \binom{n+1}{1}+ \binom{n+2}{2}+ \dots + \binom{n+k}{k}\] Using the identity \( \binom{n}{r} = \binom{n}{n-r} \), \( \binom{n}{0}+ \binom{n+1}{1}+ \binom{n+2}{2}+ \dots + \binom{n+k}{k}\) reduces to \[ \binom{n}{n}+ \binom{n+1}{n}+ \binom{n+2}{n}+ \dots + \binom{n+k}{n}\] = Coefficient of $x^n$ in $(1+x)^n$ + Coefficient of $x^n$ in $(1+x)^{n+1}$ + $\dots$ + Coefficient of $x^n$ in $(1+x)^{n+k}$ $$$$ = Coefficient of $x^n$ in \( (1+x)^n + (1+x)^{n+1} + \dots + (1+x)^{n+k} \) $$$$ = Coefficient of $x^n$ in \( (1+x)^n \frac{(1+x)^{k+1}-1}{1+x-1} = \frac{(1+x)^{n+k+1}-(1+x)^n}{x}\) $$$$ = Coefficient of $x^{n+1}$ in $(1+x)^{n+k+1}$ $= \binom{n+k+1}{n+1}$ $$$$