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# For integers n and r, let...$\left(\begin{array}{c}n\\ r\end{array}\right)=\left\{\begin{array}{ll}n{C}_{r},& ifn\ge r\ge 0\\ 0,& Otherwise\end{array}\right\$The maximum value of k for which the sum $\sum _{i=0}^{k}\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}15\\ k-i\end{array}\right)+\sum _{i=0}^{k+1}\left(\begin{array}{c}12\\ i\end{array}\right)\left(\begin{array}{c}13\\ k+1-i\end{array}\right)$ exists is equal to

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## Finding the maximum value for which the sum $\sum _{i=0}^{k}\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}15\\ k-i\end{array}\right)+\sum _{i=0}^{k+1}\left(\begin{array}{c}12\\ i\end{array}\right)\left(\begin{array}{c}13\\ k+1-i\end{array}\right)$ :Given that $\left(\begin{array}{c}n\\ r\end{array}\right)=\left\{\begin{array}{ll}n{C}_{r},& ifn\ge r\ge 0\\ 0,& Otherwise\end{array}\right\$ ${\left(1+x\right)}^{10}=10{C}_{0}+10{C}_{1}x+10{C}_{2}{x}^{2}+\dots \dots +10{C}_{10}{x}^{10}\phantom{\rule{0ex}{0ex}}{\left(1+x\right)}^{15}=15{C}_{0}+15{C}_{1}x+15{C}_{2}{x}^{2}+\dots \dots +15{C}_{k-1}{x}^{k-1}+15{C}_{k+1}{x}^{k+1}+\dots 15{C}_{15}{x}^{15}$ $\sum _{i=0}^{k}\left(10{C}_{i}\right)\left(15{C}_{k-i}\right)=10{C}_{0}15{C}_{k}+10{C}_{1}15{C}_{k-1}+....+10{C}_{k}15{C}_{0}$ Coefficient of ${x}_{k}in{\left(1+x\right)}^{25}=25{C}_{k}$ $\sum _{i=0}^{k+1}\left(12{C}_{i}\right)\left(13{C}_{k+1-i}\right)=12{C}_{0}13{C}_{k+1}+12{C}_{1}13{C}_{k}+....+12{C}_{k+1}13{C}_{0}$Coefficient of ${x}^{k+1}in{\left(1+x\right)}^{25}=25{C}_{k+1}$ $25{C}_{k}+25{C}_{k+1}=26{C}_{k+1}$For maximum valueAs per given, $k$ can be as large as possible.Hence, $k$ can be as large as possible .  Suggest Corrections  6      Similar questions
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