# Binomial Coefficient

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

## References

### 2020a

• (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Binomial_coefficient 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 $\displaystyle{ \tbinom{n}{k}. }$ 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 :

$\displaystyle{ \binom{n}{k} = \frac{n!}{k! (n-k)!}. }$

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

\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} }

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

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

$\displaystyle{ \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}. }$

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

### 2020b

• (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.