Adding combinations Show non-numerically that:
$${2\choose2} + {3\choose2} + {4\choose2} + {5\choose2} = {6\choose3}$$
The answer is as follows, but I have no idea how it was done:
$$
\begin{eqnarray}
&\phantom{=}& {2\choose2} + {3\choose2} + {4\choose2} + {5\choose2} \\
&=& {3\choose3} + {3\choose2} + {4\choose2} + {5\choose2} \\
&=& {4\choose3} + {4\choose2} + {5\choose2} \\
&=& {5\choose3} + {5\choose2} \\
&=& {6\choose3}
\end{eqnarray}
$$
Could anyone please explain this? Thank you!

If you want to choose $k$ items from $n$ items, you can do either way:


*

*choose $k$ items from the first $n-1$ items.

*choose the last item, and then choose $k-1$ items from the first $n-1$ items.
That means
${n\choose k} = {n-1\choose k} + {n-1\choose k-1}$
