Binomial Coefficients
Binomial coefficients (nCr) are the number of ways to select a set of “r” elements from “n” different elements without considering the order of arrangement of these elements (i.e., the number of unordered sets).
Binomial coefficients are also the coefficients in the expansion of ( a + b )^n, the binomial theorem.
It is believed that this formula, as well as the triangle which allows efficient calculation of the coefficients, was discovered by Blaise Pascal in the 17th century. Nevertheless, it was known to the Chinese mathematician Yang Hui, who lived in the 13th century. Perhaps it was discovered by a Persian scholar Omar Khayyam. Moreover, Indian mathematician Pingala, who lived earlier in the 3rd. BC, got similar results. The merit of Newton is that he generalized this formula for exponents that are not natural.
Calculation
Analytic formula for the calculation:
This formula can be easily deduced from the problem of ordered arrangement (number of ways to select k different elements from n different elements). First, let’s count the number of ordered selections of k elements. There are n ways to select the first element, n−1 ways to select the second element, n−2 ways to select the third element, and so on. As a result, we get the formula of the number of ordered arrangements: n(n−1)(n−2)⋯(n−k+1)=n!/(n−k)!. We can easily move to unordered arrangements, noting that each unordered arrangement corresponds to exactly k! ordered arrangements (k! is the number of possible permutations of k elements). We get the final formula by dividing n!/(n−k)! by k!.
Get Ali Qadir’s stories in your inbox
Join Medium for free to get updates from this writer.
Pascal’s Triangle
Pascal’s triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (a+ b)n, that is, the Binomial Coefficients. It is named for the 17th-century French mathematician Blaise Pascal, but it is far older.
