BinomialCoefficient.md (1493B)
1 # Binomial Coefficient 2 3 L4 4 5 **Definition:** A binomial coefficient is represented by two numbers and has a singular evaluation. The evaluation describes the number of unique subsets of the length denoted by the bottom value that can be created given a set of the length denoted by the top value. 6 7 The reason it is called the binomial coefficient is because it can be used in the expansion of binomials (ie. (x+y)^5). To use it in this case we multiply the applicable coefficient with the number of ways to select that number of a coefficient. This idea is also described as the binomial theorem. 8 9 ### Formula 10 11 (n) = n! / ((r!(n-r)!) 12 (r) 13 14 ### Example 15 16 (8) = 8! / ((3!(8-3)!)) = 40320 / (6 x 120) = 40320/720 = 56 17 (3) 18 19 8 choose 3 is 56 20 21 ### Intuition 22 23 The top of the function is all permutations of the list. The problem with this is that it includes rearrangements which we don't care for and because it is limited to the length of the entire set. 24 25 As such, we divide this by r! to account for the arrangements of the r items. The second part, (n-r!) accounts for arrangements where we are not choosing r items. 26 27 All together, we find the permutations of sets length n then divide this by r! to find the number of distinct sets not arrangements and then we divide by (n-r)! to get rid of sets that don't have r items. 28 29 ### Stats 30 31 In stats we often denote this using either the vertical denotation or the denotation $_nC_r$ where n is the length of the set and r is the size of each subset.