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$\left( {\begin{array}{*{20}{c}}{50}\\4\end{array}} \right)\,\, + \,\,\sum\limits_{i = 1}^6 {\left( {\begin{array}{*{20}{c}}{56\, - \,i}\\3\end{array}} \right)} = ......$
$\left( {\begin{array}{*{20}{c}}{55}\\4\end{array}} \right)$
$\left( {\begin{array}{*{20}{c}}{55}\\3\end{array}} \right)$
$\left( {\begin{array}{*{20}{c}}{56}\\3\end{array}} \right)$
$\left( {\begin{array}{*{20}{c}}{56}\\4\end{array}} \right)$
Solution
$\left( {\begin{array}{*{20}{c}}
{50} \\
4
\end{array}} \right)\,\, + \,\,\sum\limits_{i = 1}^6 {\left( {\begin{array}{*{20}{c}}
{56\, – \,i} \\
3
\end{array}} \right)} = \,\,\left( {\begin{array}{*{20}{c}}
{50} \\
4
\end{array}} \right)\,\, + \,\,$
$\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{54} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{53} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{52} \\
3
\end{array}} \right)\,\, + \,\left( {\begin{array}{*{20}{c}}
{51} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{50} \\
3
\end{array}} \right)$
$ = \,\,\left( {\begin{array}{*{20}{c}}
{50} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{50} \\
4
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{51} \\
3
\end{array}} \right)\,\, + $
$ + \,\,\left( {\begin{array}{*{20}{c}}
{52} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{53} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{54} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)$
$ = \,\,\left( {\begin{array}{*{20}{c}}
{51} \\
4
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{51} \\
3
\end{array}} \right)\, + \,\,\left( {\begin{array}{*{20}{c}}
{52} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{53} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{54} \\
3
\end{array}} \right)\,\, + \,\,\,\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)$
$ = \,\,\left( {\begin{array}{*{20}{c}}
{52} \\
4
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{52} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{53} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{54} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)\,\,$
$ = \,\,\,\left( {\begin{array}{*{20}{c}}
{53} \\
4
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{53} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{54} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)$
$ = \left( {\begin{array}{*{20}{c}}
{54} \\
4
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{54} \\
3
\end{array}} \right)\,\, + \,\,\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)$
$ = \left( {\begin{array}{*{20}{c}}
{55} \\
4
\end{array}} \right)\,\, + \,\left( {\begin{array}{*{20}{c}}
{55} \\
3
\end{array}} \right)\,\,\,$
$ = \,\,\left( {\begin{array}{*{20}{c}}
{56} \\
4
\end{array}} \right)$