The sum $\sum_{i = 0}^{m}{\begin{pmatrix} 10 \ i \end{pmatrix}\begin{pmatrix} 20 \ m - i \end{pmatrix},\ \ \left( where,\ \begin{pmatrix} p \ q \end{pmatrix} = 0\ if\ p

Question

The sum $\sum_{i = 0}^{m}{\begin{pmatrix} 10 \ i \end{pmatrix}\begin{pmatrix} 20 \ m - i \end{pmatrix},\ \ \left( where,\ \begin{pmatrix} p \ q \end{pmatrix} = 0\ if\ p < q ight)}$ is maximum, when $m$ is

Dr. Abubacker · Accepted Answer

(c)
$\because\ \sum_{i = 0}^{m}{\begin{pmatrix}
10 \
i
\end{pmatrix}\begin{pmatrix}
20 \
m - i
\end{pmatrix} = \ \sum_{i = 0}^{m}{\ ^{10}C_{i} \bullet^{20}C_{m - 1}}\ }$
$= Coefficient\ of\ x^{m}\ in\ the\ expansion\ of\ {(1 + x)}^{10}{(1 + x)}^{20}$
$=^{30}C_{m}$
It is maximum, when
$m = \frac{30}{2} = 15$

Q. The sum $\sum_{i = 0}^{m}{\begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 20 \\ m - i \end{pmatrix},\ \ \left( where,\ \begin{pmatrix} p \\ q \end{pmatrix} = 0\ if\ p < q \right)}$ is maximum, when $m$ is

Explanation

Reviewed by Dr. Abubacker

Want more practice?