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$\left( {\,_{\,8}^{15}\,} \right) + \left( {\,_{\,9}^{15}\,} \right) - \left( {\,_{\,6}^{15}\,} \right) - \left( {\,_{\,7}^{15}\,} \right) = ......$
$0$
$1$
$2$
$3$
Solution
$\left( {\begin{array}{*{20}{c}}
{15} \\
8
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{15} \\
9
\end{array}} \right) – \left[ {\left( {\begin{array}{*{20}{c}}
{15} \\
6
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{15} \\
7
\end{array}} \right)} \right]$
$ = \left( {\begin{array}{*{20}{c}}
{16} \\
9
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{16} \\
7
\end{array}} \right)\,\,\,\left[ {\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)} \right]$
$ = \left( {\begin{array}{*{20}{c}}
{16} \\
9
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{16} \\
{16 – 7}
\end{array}} \right)\,\,\,\left[ {\because \left( {\begin{array}{*{20}{c}}
n \\
r
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
n \\
{r – 1}
\end{array}} \right)} \right]\,\,$
$ = \left( {\begin{array}{*{20}{c}}
{16} \\
9
\end{array}} \right) – \left( {\begin{array}{*{20}{c}}
{16} \\
9
\end{array}} \right) = 0$
