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$\left( {_{\,1}^{10}} \right) + \left( {_{\,2}^{10}} \right) + \left( {_{\,3}^{11}} \right) + \left( {_{\,4}^{12}} \right) + \left( {_{\,5}^{13}} \right) = ...........$
$\left( {_{\,6}^{14}} \right)$
$\left( {_{\,7}^{13}} \right)$
$\left( {_{\,6}^{13}} \right)$
$\left( {_{\,5}^{14}} \right)$
Solution
$\left( {\begin{array}{*{20}{c}}
n \\
r
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
n \\
{r – 1}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{n + 1} \\
r
\end{array}} \right)\,\,$
$\left( {\begin{array}{*{20}{c}}
{10} \\
1
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{10} \\
2
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{11} \\
3
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{12} \\
4
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{13} \\
5
\end{array}} \right)$
$ = \left( {\begin{array}{*{20}{c}}
{11} \\
2
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{11} \\
3
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{12} \\
4
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{13} \\
5
\end{array}} \right)\,\,$
$ = \left( {\begin{array}{*{20}{c}}
{12} \\
3
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{12} \\
4
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{13} \\
5
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{13} \\
4
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{13} \\
5
\end{array}} \right)$
$ = \left( {\begin{array}{*{20}{c}}
{14} \\
5
\end{array}} \right)$ મળે
