Question

For integers $n$ and $r,$ let $\left(\begin{array}{c}n\\ r\end{array}\right)=\{\begin{array}{cc}{}^n{C}_r,& \text{if}n\ge r\ge 0\\ 0,& \text{otherwise}\end{array}$

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

BONUS QUESTION :

$(1+x{)}^{10}={}^{10}{C}_0+{}^{10}{C}_{1}x+{}^{10}{C}_2{x}^2+\cdots +{}^{10}{C}_{10}{x}^{10}$
$(1+x{)}^{15}={}^{15}{C}_0+{}^{15}{C}_{1}x+\cdots +{}^{15}{C}_{k-1}{x}^{k-1}+{}^{15}{C}_k{x}^k+{}^{15}{C}_{k+1}{x}^{k+1}+\cdots +{}^{15}{C}_{15}{x}^{15}$
$\sum _{i=0}^k{(}^{10}{C}_i\left){(}^{15}{C}_{k-i}\right)={}^{10}{C}_0\cdot {}^{15}{C}_k+{}^{10}{C}_1\cdot {}^{15}{C}_{k-1}+\cdots +{}^{10}{C}_k\cdot {}^{15}{C}_0$
Coefficient of $x^k$ in $(1+x{)}^{25}$ $={}^{25}{C}_k$

$\sum _{i=0}^{k+1}{(}^{12}{C}_i\left){(}^{13}{C}_{k+1-i}\right)={}^{12}{C}_0\cdot {}^{13}{C}_{k+1}+{}^{12}{C}_1\cdot {}^{13}{C}_k+\cdots +{}^{12}{C}_{k+1}\cdot {}^{13}{C}_0$
Coefficient of $x^{k+1}$ in $(1+x{)}^{25}$ $={}^{25}{C}_{k+1}$


$\therefore \sum _{i=0}^k\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}15\\ k-i\end{array}\right)+\sum _{i=0}^{k+1}\left(\begin{array}{c}12\\ i\end{array}\right)\left(\begin{array}{c}13\\ k+1-i\end{array}\right)$
$={}^{25}{C}_k+{}^{25}{C}_{k+1}$
$={}^{26}{C}_{k+1}$
By the given definition of $\left(\begin{array}{c}n\\ r\end{array}\right),$ $k$ can be as large as possible.