Permutations and Combinations

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.