Manuals/calci/BINOMIALCOEFFICIENT

BINOMIAL(N,K)

Description

This function gives the coefficent of the binomial distribution.
Binomial coefficient is the set of positive integer which equals the number of combinations of k items that can be selected from a set of n items.
The coefficients satisfy the Pascals recurrence.
The binomial coefficents are denoted by ${\binom {n}{k}}$ and it is read by n choose k.
It is the coefficient of the $x^{k}$ term in the polynomial expansion of the binomial thorem $(1+x)^{n}$ .
The coefficient is occur in the formula of binomial thorem:

 $(x+y)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}y^{k}$  where  $k\leq n$ .

  1. Recursive formula 
  2. Multiplicative formula 
  3. Factorial formula.

 ${\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}}$   for  $n,k>0$  and  $1\leq k\leq n-1$ .

${\binom {n}{k}}=\prod _{i=1}^{k}{\frac {n+1-i}{i}}$

${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}$ where $k\leq n$ ,and which is zero when $k>n$ .

Also for the initial values ${\binom {n}{0}}={\binom {n}{n}}=1$ for $n\geq 0$ .
Most compact formula for the coefficient of the binomial value is Factorial formula.
Factorial formula is symmetric of the combination formula.