binomial coefficients, positive integers that are the numerical coefficients of the binomial theorem, which expresses the expansion of (a + b)ⁿ. The nth power of the sum of two numbers a and b may be expressed as the sum of n + 1 terms of the form

in the sequence of terms, the index r takes on the successive values 0, 1, 2,…, n. The binomial coefficients are defined by the formula

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combinatorics: Binomial coefficients

in which n! (called n factorial) is the product of the first n natural numbers 1, 2, 3,…, n (and where 0! is defined as equal to 1).

The coefficients may also be found in the array often called Pascal’s triangle

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by finding the rth entry of the nth row (counting begins with a zero in both directions). Each entry in the interior of Pascal’s triangle is the sum of the two entries above it. Thus, the powers of (a + b)ⁿ are 1, for n = 0; a + b, for n = 1; a² + 2ab + b², for n = 2; a³ + 3a²b + 3ab² + b³, for n = 3; a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴, for n = 4, and so on. (Although it is called “Pascal’s triangle,” this array was known to Islamic and Chinese mathematicians of the late medieval period. Al-Karajī calculated Pascal’s triangle about 1000 ce, and Jia Xian calculated Pascal’s triangle up to n = 6 in the mid-11th century.)

In addition, the binomial coefficients appear in probability and combinatorics as the number of combinations that a set of k objects selected from a set of n objects can produce without regard to order. The number of such subsets is denoted by _nC_k, read “n choose k,” with the following combination formula:

This is the same as the binomial coefficient of the kth term of (a+b)ⁿ. For example, the number of combinations of five objects taken two at a time is