(IME - 2021/2022)Considere as propriedades dos coe

$\begin{pmatrix}100\\0 \end{pmatrix}^2+\begin{pmatrix}100\\1 \end{pmatrix}^2+...+\begin{pmatrix}100\\100 \end{pmatrix}^2=\begin{pmatrix}200\\100 \end{pmatrix}^2$.

$\begin{pmatrix}100\\39 \end{pmatrix}+\begin{pmatrix}100\\40 \end{pmatrix}=\begin{pmatrix}101\\40 \end{pmatrix}$.

$2\times1\times \binom{100}{2}+3\times2\times\binom{100}{3}+4\times3\times\binom{100}{4}+...+100\times99\times\binom{100}{100}=9900\times2^{98}$.

$\binom{100}{1}+2\times\binom{100}{2}+3\times\binom{100}{3}+...+100\times\binom{100}{100}=100\times2^{99}$

$1-\binom{100}{1}+\binom{100}{2}-\binom{100}{3}+...+\binom{100}{100}=0$