A) 5
B) 15
C) 10
D) 20
Correct Answer: B
Solution :
For m \[=5,\,\,\,\,\sum\limits_{i=0}^{5}{\left( \begin{align} & \,10\, \\ & \,\,i \\ \end{align} \right)\,\left( \begin{align} & \,\,20\, \\ & 5-i\, \\ \end{align} \right)}\] \[=\left( \begin{align} & \,10 \\ & \,\,0 \\ \end{align} \right)\,\,\left( \begin{align} & 20 \\ & \,5 \\ \end{align} \right)\,\,+\,\left( \begin{align} & 10 \\ & \,1 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,4 \\ \end{align} \right)\,+...+\left( \begin{align} & 10 \\ & \,5 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,0 \\ \end{align} \right),\] for m = 10, \[\sum\limits_{i=0}^{10}{\left( \begin{align} & \,10\, \\ & \,\,i \\ \end{align} \right)\,\left( \begin{align} & \,\,20\, \\ & 10-i\, \\ \end{align} \right)}\] \[=\left( \begin{align} & 10 \\ & \,0 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & 10 \\ \end{align} \right)+\left( \begin{align} & 10 \\ & \,1 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,9\, \\ \end{align} \right)+\left( \begin{align} & 10 \\ & \,2 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,8\, \\ \end{align} \right)\]\[+...+\left( \begin{align} & 10\, \\ & 10 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,0 \\ \end{align} \right)\], for m = 15, \[\sum\limits_{i=0}^{15}{\left( \begin{align} & \,10\, \\ & \,\,i \\ \end{align} \right)\,\left( \begin{align} & \,\,\,20\, \\ & 15-i\, \\ \end{align} \right)}\] \[=\left( \begin{align} & \,10 \\ & \,\,0 \\ \end{align} \right)\,\,\left( \begin{align} & 20 \\ & \,15 \\ \end{align} \right)\,\,+\,\left( \begin{align} & 10 \\ & \,1 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & 1\,4 \\ \end{align} \right)\,+\left( \begin{align} & 10 \\ & \,2 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,13 \\ \end{align} \right)+..+\left( \begin{matrix} 10 \\ 10 \\ \end{matrix} \right)\,\left( \begin{matrix} 20 \\ 5 \\ \end{matrix} \right)\] and for m = 20, \[\sum\limits_{i=0}^{20}{\left( \begin{align} & \,10\, \\ & \,\,i \\ \end{align} \right)\,\left( \begin{align} & \,\,\,20\, \\ & 20-i\, \\ \end{align} \right)}\] \[=\left( \begin{align} & \,10 \\ & \,\,0 \\ \end{align} \right)\,\,\left( \begin{align} & 20 \\ & 20 \\ \end{align} \right)\,\,+\,\left( \begin{align} & 10 \\ & \,1 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & 1\,9 \\ \end{align} \right)\,+...+\left( \begin{align} & 10 \\ & 10 \\ \end{align} \right)\,\left( \begin{align} & 20 \\ & \,10 \\ \end{align} \right)\] Clearly, the sum is maximum for m = 15. Note that \[^{10}{{C}_{r}}\] is maximum for r = 5 and \[^{20}{{C}_{r}}\] is maximum for r = 10. Note that the single term \[^{10}{{C}_{5}}\,\times {{\,}^{20}}{{C}_{10}}\,\](in case m = 15) is greater than the sum \[^{10}{{C}_{0}}{{\,}^{20}}{{C}_{10}}\,+{{\,}^{10}}{{C}_{1}}{{\,}^{20}}{{C}_{9}}\,+{{\,}^{10}}{{C}_{2}}{{\,}^{20}}{{C}_{8}}\,+.....\] \[^{10}{{C}_{8}}{{\,}^{20}}{{C}_{2}}\,+{{\,}^{10}}{{C}_{9}}{{\,}^{20}}{{C}_{1}}\,+{{\,}^{10}}{{C}_{10}}{{\,}^{20}}{{C}_{0}}\](in case m = 10). Also the sum in case m = 10 is same as that in case m = 20.
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