Question

The value of $\begin{pmatrix}50 \\ 0\end{pmatrix}\begin{pmatrix}50 \\ 1\end{pmatrix}+\begin{pmatrix}50 \\ 1\end{pmatrix}\begin{pmatrix}50 \\ 2\end{pmatrix}+\ldots \ldots \ldots+\begin{pmatrix}50 \\ 49\end{pmatrix}\begin{pmatrix}50 \\ 50\end{pmatrix}$ is, where ${ }^n C_r=\begin{pmatrix}n \\ r\end{pmatrix}$

• A. $\begin{pmatrix}100 \\ 49\end{pmatrix}$
• B. $\begin{pmatrix}100 \\ 26\end{pmatrix}$
• C. $\begin{pmatrix}50 \\ 25\end{pmatrix}$
• D. $\begin{pmatrix}50 \\ 25\end{pmatrix}^2$

Answer 1

Answer: (A)

${ }^{50} C _0 \times{ }^{50} C _1+{ }^{50} C \times \times{ }^{50} C _2+\ldots \ldots \ldots . .+{ }^{50} C _{49} \times{ }^{50} C _{50}$
$={ }^{50} C _0 \times{ }^{50} C _{49}+{ }^{50} C _1 \times{ }^{50} C _{48}+\ldots \ldots \ldots \ldots .+{ }^{50} C _{49} \times{ }^{50} C _0$
$=$ co-eff. of $x ^{49}$ in $(1+ x )^{100}={ }^{100} C _{49}$