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Summary
Description5-choose-3 tree proof.svg
English: Start at the center, with the understanding that 5-choose-3 three is unknown (pretend you can't count the dotted lines.) Letting 5-choose3 equal X include subscripts on 2 Rs and 3 Cs to conclude that their permutations are 2! and 3!, respectively. After both letters have been subscripted and all permutations are counted, we use the product rule to conclude that the outer branches of the tree has (2!)(3!)(X) endpoints. But we also know that this equals the permutation of 5 objects. Hence 5!=(2!)(3!)(X). Solving yields X=10.
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