gen-in-3-4-sentences-explain-double-counting-state-any-key-formu-789f

In 3-4 sentences, explain double counting. State any key formula.

# Double Counting Double counting is a combinatorial proof technique where you count the same set of objects in two different ways and equate the results. By establishing that both counting methods enumerate the same collection, you can derive identities and solve problems that might be difficult using direct approaches. **Key formula pattern:** If sets A and B can be counted in two ways, then: $$\sum_{a \in A} f(a) = \sum_{b \in B} g(b)$$ A classic example is proving $\binom{n}{k} = \binom{n}{n-k}$ by counting subsets of size $k$ versus counting their complements of size $n-k$.