Binomial theorem

Japanese: 二項定理 - にこうていり

The formula for the expansion of the algebraic sum of two terms to the nth power ( n is a positive integer) is called the binomial theorem. In formula form,

Each coefficient

are called binomial coefficients, and are sometimes represented by symbols. The value of _n C _k is the total number of ways to select k items from n different items (the order of selection is not taken into account) (the number of combinations). The binomial theorem is related to combinatorics, and the properties of binomial coefficients were clarified by Pascal. One of the most famous examples is Pascal's triangle, which is the relationship between binomial coefficients , _n C _{k -1} + _n C _k = _{n +1} C _k (1≦ k ≦n).
The numbers are arranged in a triangular shape.

The expansion formula in which the positive integer exponent n in the binomial theorem is replaced by a general real number α is called the general binomial theorem. In this case, the number of expansion terms becomes infinite. For example,

The right-hand side is called the binomial series of x . The range of possible values of x for this series to have a sum (converge) is -1≦ x ≦1.

In general, when (1 + x ) ^α is expanded into a binomial series, the convergence domain is -1≦ x ≦1 if α＞0, and -1＜ x ≦1 if -1＜ α＜0. If α=0 or α is a natural number, the series converges for any number of x . General binomial series were studied by Isaac Newton.

[Yoshio Takeuchi]

Pascal's Triangle
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Pascal's Triangle

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

二項の代数和のn乗（nは正整数）の展開公式を二項定理という。式で書けば、

となる。各係数

を二項係数といい、これをまた記号で表すこともある。この_nC_kの数値はn個の異なるものからk個のものを選び出す仕方（選出の順序は考えない）の総数である（組合せの数）。二項定理は組合せ理論と関係し、二項係数についての性質はパスカルによって明らかにされた。そのなかの有名なものにパスカルの三角形がある。これは二項係数間の関係
　　_nC_k-1+_nC_k=_n+1C_k (1≦k≦n)
を三角形状に数を並べて示したものである。