The value $r$ for which $\left ( \overset{30}{r} \right )\left ( \overset{15}{r} \right )+\left ( \overset{30}{r-1} \right )\left ( \overset{15}{1} \right ) +..........+\left ( \overset{15}{r} \right )\left ( \overset{30}{r} \right )$ is maximum is/are

Consider the following series.

$\begin{array}{l} { }^{m} C_{0}{ }^{n} C_{r}+{ }^{m} C_{1}{ }^{n} C_{r-1}+\ldots{ }^{m} C_{r}{ }^{n}{ }^{n} C_{0} \\ =\text { Coefficient of } \mathrm{x}^{r} \text { in }{ }_{(1+x)^{m} \cdot(x+1)^{n}} \\ =\text { coefficient of } x^{r} \text { in }(1+\mathrm{x})^{m+n} \\ ={ }^{m} n c_{r} \\ \end{array}$

Here $\mathrm{m}=30 \mathrm{n}=15$.

Hence ther series turns out to be

$\begin{array}{c} { }^{30+15} \mathrm{C}_{\mathrm{r}} \\ \mathrm{T}_{\mathrm{r}+1}={ }^{45} \mathrm{C}_{\mathrm{r}} \end{array}$

Now it is maximum when $T_{r+1}$ is the middle term.

Hence The middle terms are $24^{\text {th }}$ and $23^{\text {rd }}$ terms. Hence

$r=22 \text { or } r=23 \text {. }$