Combinations (nCr)

How to read it

OpenStax’s Contemporary Mathematics reads 5C3 as “the number of combinations of 5 objects taken 3 at a time” (ten ways to pick 3 letters from A to E) and gives the formula nCr = n!/(r!(n − r)!); 8C3 = 56. When the order does matter, the count is the number of permutations, nPr = n!/(n − r)!, which is larger by a factor of r!, the number of ways to arrange the r chosen things. Oscar Levin’s Discrete Mathematics writes permutations as P(n, k), and Grinstead and Snell’s Introduction to Probability (Dartmouth) define the combination count as the number of subsets with j elements that can be chosen from a set of n.

Also searched as: binomial coefficient, combination formula, n choose k, n choose r, nCr, nCr meaning, nPr, permutations symbol.

nCr, C(n, r) or n over r

The same number has several names. In the bracket form, n above r in tall round brackets, it is read “n choose r” and called a binomial coefficient (Levin; Grinstead and Snell), because the same numbers are the coefficients when a binomial such as (x + y)ⁿ is multiplied out. OpenStax writes nCr with small letters before and after the C. Permutations have matching names, nPr in OpenStax and P(n, k) in Levin. All count selections; P counts the orders as different, C doesn’t.