Combinatorics

$\sum_{i=2}^{99} {(-1)^i}\frac{99!}{i!(99-i)!} =$ $\sum_{i=0}^{99} {(-1)^i}\frac{99!}{i!(99-i)!} - {(-1)^1}\frac{99!}{1!(99-1)!} - {(-1)^0}\frac{99!}{0!(99-0)!}=$ $\sum_{i=0}^{99} {(1)^{99-i}(-1)^i}\frac{99!}{i!(99-i)!} + 99 -1 =$ $[1+(-1)]^{99}+98 =$ $0+98 =98$

$\begin{aligned} S & = \sum_{k=2}^{99} \frac {(-1)^k99!}{k!(99-k!)} \\ & = {\color{#3D99F6} \frac {99!}{2!97!}} - {\color{#D61F06} \frac {99!}{3!96!}} + {\color{#20A900} \frac {99!}{4!95!}} - \cdots + {\color{#69047E} \frac {99!}{44!45!}} - {\color{#69047E} \frac {99!}{45!44!}} + \cdots - {\color{#20A900} \frac {99!}{95!4!}} + {\color{#D61F06} \frac {99!}{96!3!}} - {\color{#3D99F6} \frac {99!}{97!2!}} + \frac {99!}{98!1!} - \frac {99!}{99!0!} \\ & = {\color{#3D99F6} \cancel{\frac {99!}{2!97!}}} - {\color{#D61F06} \cancel{\frac {99!}{3!96!}}} + {\color{#20A900} \cancel{\frac {99!}{4!95!}}} - \cdots + {\color{#69047E} \cancel{\frac {99!}{44!45!}}} - {\color{#69047E} \cancel{\frac {99!}{45!44!}}} + \cdots - {\color{#20A900} \cancel{\frac {99!}{95!4!}}} + {\color{#D61F06} \cancel{\frac {99!}{96!3!}}} - {\color{#3D99F6} \cancel{\frac {99!}{97!2!}}} + 99 - 1 \\ & = \boxed{98} \end{aligned}$