First slide

Binomial theorem for positive integral Index

Question

The sum $\sum_{i = 0}^{m} (\begin{matrix} 10 \\ i \end{matrix}) (\begin{matrix} 20 \\ m - i \end{matrix}), [where, (\begin{array}{l} p \\ q \end{array}) = 0, if p < q]$ is maximum, when m is

Moderate

A

5
B

10
C

15
D

20

Solution

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

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