Coefficient of $x^{99}$ in $x^{100}+2x^{99}(1+x)+3(1+x)^2x^{98}+\cdots+101(1+x)^{100}$ is
$^{101}C_{99}-\,^{101}C_{98}$
$100(^{101}C_{99})-\,^{101}C_{98}$
$-\,^{101}C_{98}+101(^{101}C_{99})$
$101(^{101}C_{98})-\,^{101}C_{99}$
Step-by-Step Solution
Key Concept: General term is $(r+1)(1+x)^r x^{100-r}$; coefficient of $x^{99}$ in each term
Coeff of $x^{99}$ in $(r+1)(1+x)^r x^{100-r}$: need power of $x$ from $(1+x)^r$ = $(r-1)$. Contribution = $(r+1)\binom{r}{r-1}$... Sum $= 101\cdot\binom{101}{99}-\binom{101}{98}$.
Correct Answer: 3