“binomial coefficient”的概念、定义、翻译、参考文献-科学参考

binomial coefficient

Mathematics

The number, denoted by $(\begin{matrix} n \\ r \end{matrix})$ , where n is a positive integer and r is an integer such that 0 ≤ r ≤ n, defined by the formula

$(\begin{matrix} n \\ r \end{matrix}) = \frac{n (n - 1) … (n - r + 1)}{1 \times 2 \times \dots \times r} = \frac{n!}{r! (n - r)!} .$

Since, by convention, 0! = 1, we have $(\begin{matrix} n \\ 0 \end{matrix}) = (\begin{matrix} n \\ n \end{matrix}) = 1 .$ These numbers are called binomial coefficients because they occur as coefficients in the Binomial Theorem. They are sometimes denoted by ⁿC_r; this notation arose as the binomial coefficient equals the number of ways of selecting r objects out of n (see selection). The numbers have the following properties:

$\begin{array}{l} (i) (\begin{matrix} n \\ r \end{matrix}) is an integer (this is not obvious \\ from the definition). \\ (ii) (\begin{matrix} n \\ n - r \end{matrix}) = (\begin{matrix} n \\ r \end{matrix}) . \\ (iii) (\begin{matrix} n + 1 \\ r \end{matrix}) = (\begin{matrix} n \\ r - 1 \end{matrix}) + (\begin{matrix} n \\ r \end{matrix}) . \\ (iv) (\begin{matrix} n \\ 0 \end{matrix}) + (\begin{matrix} n \\ 1 \end{matrix}) + (\begin{matrix} n \\ 2 \end{matrix}) + \dots + (\begin{matrix} n \\ n \end{matrix}) = 2^{n} . \end{array}$

It is instructive to see the binomial coefficients laid out in the form of Pascal’s triangle.

Statistics

The coefficient of a power of x in the binomial expansion of (1+x)ⁿ. The general binomial coefficient is defined, for any value of n and for any positive integer r, by $binomial coefficient$ By convention, . Suppose now that n is a non-negative integer. In this case is an integer and . Furthermore, $binomial coefficient$ If r≤n then is also written as ⁿC_r (see combination) and $binomial coefficient$ The phrase ‘binomial coefficient’ appears in English in a mathematics text of 1876.

单词	binomial coefficient
释义	binomial coefficient Mathematics The number, denoted by $(\begin{matrix} n \\ r \end{matrix})$ , where n is a positive integer and r is an integer such that 0 ≤ r ≤ n, defined by the formula $(\begin{matrix} n \\ r \end{matrix}) = \frac{n (n - 1) … (n - r + 1)}{1 \times 2 \times \dots \times r} = \frac{n!}{r! (n - r)!} .$ Since, by convention, 0! = 1, we have $(\begin{matrix} n \\ 0 \end{matrix}) = (\begin{matrix} n \\ n \end{matrix}) = 1 .$ These numbers are called binomial coefficients because they occur as coefficients in the Binomial Theorem. They are sometimes denoted by ⁿC_r; this notation arose as the binomial coefficient equals the number of ways of selecting r objects out of n (see selection). The numbers have the following properties: $\begin{array}{l} (i) (\begin{matrix} n \\ r \end{matrix}) is an integer (this is not obvious \\ from the definition). \\ (ii) (\begin{matrix} n \\ n - r \end{matrix}) = (\begin{matrix} n \\ r \end{matrix}) . \\ (iii) (\begin{matrix} n + 1 \\ r \end{matrix}) = (\begin{matrix} n \\ r - 1 \end{matrix}) + (\begin{matrix} n \\ r \end{matrix}) . \\ (iv) (\begin{matrix} n \\ 0 \end{matrix}) + (\begin{matrix} n \\ 1 \end{matrix}) + (\begin{matrix} n \\ 2 \end{matrix}) + \dots + (\begin{matrix} n \\ n \end{matrix}) = 2^{n} . \end{array}$ It is instructive to see the binomial coefficients laid out in the form of Pascal’s triangle. Statistics The coefficient of a power of x in the binomial expansion of (1+x)ⁿ. The general binomial coefficient is defined, for any value of n and for any positive integer r, by $binomial coefficient$ By convention, . Suppose now that n is a non-negative integer. In this case is an integer and . Furthermore, $binomial coefficient$ If r≤n then is also written as ⁿC_r (see combination) and $binomial coefficient$ The phrase ‘binomial coefficient’ appears in English in a mathematics text of 1876.
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