Combinations Formula

Algebra 2 · Axiom Academy

LESSON The Combinations Formula Counting selections where order doesn't matter — deriving . A combination is a selection in which the order is irrelevant . Rearranging the same items gives the same combination, so and are one selection, not two. All six orderings describe one selection 2. From Permutations to Combinations If order did matter we would count permutations . Choosing and arranging 2 of the five letters gives ordered pairs. But each unordered pair — say — is counted 2! = 2 times, once as AB and once as BA . AB and BA are different · _5P_2 = 20 AB and BA are the same · _5C_2 = 10 Replacing _nP_r with and dividing by r! assembles the whole formula. Each factorial earns its place: n! arranges everything, r! cancels the reorderings of the r chosen items, and (n-r)! cancels the reorderings of the rest. 4. Symmetry, and a Worked Count Choosing which r items to include automatically decides which n-r items to leave out — so the number of ways to pick r equals the number of ways to pick n-r . The denominators and are identical — which is exactly why _5C_2 = _5C_3 . The same symmetry gives, for instance: You can now count unordered selections: a combination ignores order, and counts them. Scroll up to replay any animation.

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