The value of $r$ for which $S =^{20} C_{r}^{10} C_{0} +^{20} C_{r - 1}^{10} C_{1} . . . . . +^{20} C_{0}^{10} C_{r}$ is maximum is

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Solution

The correct option is D 15

$S =^{20} C_{r}^{10} C_{0} +^{20} C_{r - 1}^{10} C_{1} + \dots +^{20} C_{0}^{10} C_{r}$

This is the sum of coefficients of $x^{r}$ in $(1 + x)^{20} (1 + x)^{10} \Rightarrow (1 + x)^{30}$
So coefficient of $x^{r} =^{30} C_{r}$
For $r =$ maximum $, r = \frac{30}{2} = 15$

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