Question

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

Answer 1

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 $\left(\begin{array}{c}n\\ r\end{array}\right)=\left\{\begin{array}{ll}n{C}_r,& ifn\ge r\ge 0\\ 0,& Otherwise\end{array}\right.$

$$\begin{gathered} {(1+x)}^{10}=10C_0+10C_{1}x+10C_2{x}^2+\dots \dots +10C_{10}{x}^{10} \\ {(1+x)}^{15}=15C_0+15C_{1}x+15C_2{x}^2+\dots \dots +15C_{k-1}{x}^{k-1}+15C_{k+1}{x}^{k+1}+\dots 15C_{15}{x}^{15} \end{gathered}$$ $\sum _{i=0}^k\left(10C_i\right)\left(15C_{k-i}\right)=10C_{0}15C_k+10C_{1}15C_{k-1}+....+10C_{k}15C_0$

Coefficient of $x_{k}in{(1+x)}^{25}=25C_k$

$\sum _{i=0}^{k+1}\left(12C_i\right)\left(13C_{k+1-i}\right)=12C_{0}13C_{k+1}+12C_{1}13C_k+....+12C_{k+1}13C_0$

Coefficient of $x^{k+1}in{(1+x)}^{25}=25C_{k+1}$

$25C_k+25C_{k+1}=26C_{k+1}$

For maximum value

As per given, $k$ can be as large as possible.

Hence, $k$ can be as large as possible .