Number of ways a subset of objects can be selected from a given set of objects. In a permutation, order is important; in a combination, it is not. Thus, there are six permutations of the letters A, B, C selected two at a time (AB, AC, BC, BA, CA, CB) yet only three combinations (AB, AC, BC). The number of permutations of math.r objects chosen from a set of math.n objects, expressed in factorial notation, is math.n! ÷ (math.n − math.r)! The number of combinations is math.n! ÷ [math.r!(math.n − math.r)!]. The (math.r + 1)st coefficient in the binomial expansion of (math.x + math.y)^math.n coincides with the combination of math.n objects chosen math.r at a time (see binomial theorem). Probability theory evolved from the study of gambling, including figuring out combinations of playing cards or permutations of win-place-show possibilities in a horse race, and such counting methods played an important role in its development in the 17th century.

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