Binomial Coefficient

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A Binomial Coefficient is a Coefficient that is a positive integer in the binomial theorem.



  • (Wikipedia, 2020) ⇒ Retrieved:2020-9-13.
    • In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers nk ≥ 0 and is written [math]\displaystyle{ \tbinom{n}{k}. }[/math] It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula :

      [math]\displaystyle{ \binom{n}{k} = \frac{n!}{k! (n-k)!}. }[/math]

      For example, the fourth power of 1 + x is :

      [math]\displaystyle{ \begin{align} (1 + x)^4 &= \tbinom{4}{0} x^0 + \tbinom{4}{1} x^1 + \tbinom{4}{2} x^2 + \tbinom{4}{3} x^3 + \tbinom{4}{4} x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end{align} }[/math]

      and the binomial coefficient[math]\displaystyle{ \tbinom{4}{2} =\tfrac{4!}{2!2!} = 6 }[/math] is the coefficient of the x2 term.

      Arranging the numbers [math]\displaystyle{ \tbinom{n}{0}, \tbinom{n}{1}, \ldots, \tbinom{n}{n} }[/math] in successive rows for [math]\displaystyle{ n=0,1,2,\ldots }[/math] gives a triangular array called Pascal's triangle, satisfying the recurrence relation :

      [math]\displaystyle{ \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}. }[/math]

      The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol [math]\displaystyle{ \tbinom{n}{k} }[/math] is usually read as "n choose k" because there are [math]\displaystyle{ \tbinom{n}{k} }[/math] ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are [math]\displaystyle{ \tbinom{4}{2}=6 }[/math] ways to choose 2 elements from [math]\displaystyle{ \{1,2,3,4\}, }[/math] namely [math]\displaystyle{ \{1,2\} \text{, } \{1,3\} \text{, } \{1,4\} \text{, } \{2,3\} \text{, } \{2,4\} \text{,} }[/math] and [math]\displaystyle{ \{3,4\}. }[/math] The binomial coefficients can be generalized to [math]\displaystyle{ \tbinom{z}{k} }[/math] for any complex number and integer k ≥ 0, and many of their properties continue to hold in this more general form.


  • (MathWorld, 2020) ⇒ Weisstein, Eric W. "Binomial Coefficient". From: MathWorld--A Wolfram Web Resource. Retrieved:2020-9-13.
    • QUOTE: The binomial coefficient $\left(\begin{array}{c} n \\ k \end{array}\right)$ is the number of ways of picking $k$ unordered outcomes from $n$ possibilities, also known as a combination or combinatorial number. The symbols $n$$C_k$ and $\left(\begin{array}{c} n \\ k \end{array}\right)$ are used to denote a binomial coefficient, and are sometimes read as "$n$ choose $k$." $\left(\begin{array}{c} n \\ k \end{array}\right)$ therefore gives the number of $k$-subsets possible out of a set of $n$ distinct items. For example, The 2-subsets of $\{1,2,3,4\}$ are the six pairs $\{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}$, and $\{3,4\}$, so $\left(\begin{array}{c} 4 \\ 2 \end{array}\right)=6$. The number of lattice paths from the origin $(0,0)$ to a point $(a,b)$ is the binomial coefficient $\left(\begin{array}{c} a+b \\ a \end{array}\right)$ (Hilton and Pedersen 1991).

      The value of the binomial coefficient for nonnegative $n$ and $k$ is given explicitly by

      ${ }_{n} C_{k} \equiv\left(\begin{array}{c}n \\k\end{array}\right) \equiv \frac{n !}{(n-k) ! k !}$

      where $z!$ denotes a factorial.