Some Random Short Tricks

Choose one, leave one

\[\binom{n}{k}= \binom{n-1}{k-1} + \binom{n-1}{k}\]

This could be thought of as choosing one item and leaving one item while choosing \(k\) items from \(n\) items.

Choose one item: We have \(\binom{n-1}{k-1}\) ways to choose remaining \(k-1\) items from \(n-1\) items Leave one item: We have \(\binom{n-1}{k}\) ways to choose \(k\) items from remaining \(n-1\) items.