Manuals/calci/BINOMIAL

BINOMIAL(n,k)

• $n$ is the number of items.
• $k$ is the number of selection.

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 $k\leq n$. 

• To find the coefficient of the binomial ,we can use several methods.

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

• 1.Recursive Formula:

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

• 2. Multiplicative formula:

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

• 3.Factorial formula:

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

Examples

1. =BINOMIAL(10,3)= 120
2. =BINOMIAL(20,7)= 77520
3. =BINOMIAL(15,0)= 1
4. =BINOMIAL(12,12)=1
5. =BINOMIAL(1,-1) = 0

See Also

• BINOMDIST
• BINOMIALDISTRIBUTED