# Manuals/calci/BINOMIAL

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**BINOMIAL(n,k)**

- is the number of items.
- 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 and it is read by n choose k.
- It is the coefficient of the term in the polynomial expansion of the binomial thorem .
- The coefficient is occur in the formula of binomial thorem:

where .

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

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

- 1.Recursive Formula:

for and .

- 2. Multiplicative formula:

- 3.Factorial formula:

where ,and which is zero when .

- Also for the initial values for .
- Most compact formula for the coefficient of the binomial value is Factorial formula.
- Factorial formula is symmetric of the combination formula.

## ZOS

- The syntax is to calculate BINOMIAL in ZOS is .
- is the number of items.
- is the number of selection.

- For e.g., BINOMIAL(20..25,4)
- BINOMIAL(10..14,7..8)

## Examples

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