Manuals/calci/BINOMIAL

BINOMIAL(N,K)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the number of items.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 + x)^n} .
• The coefficient is occur in the formula of binomial thorem:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x+y)^n=\sum _{k=0}^n \binom{n}{k} x^{n-k} y^k}
 where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  k\le 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{n}{k}= \binom{n-1}{k-1} +\binom{n-1}{k}}
  for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n,k>0}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\le k\le n-1}
.

• 2. Multiplicative formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{n}{k}= \prod_{i=1}^k \frac{n+1-i}{i}}

• 3.Factorial formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{n}{k}= \frac{n!}{k!(n-k)!}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\le n} ,and which is zero when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k>n} .

• Also for the initial values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{n}{0}=\binom{n}{n}=1 } for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ge 0} .
• 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BINOMIAL (N,K)} .
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the number of items.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is the number of selection.
• For e.g., BINOMIAL(20..25,4)
• BINOMIAL(10..14,7..8)

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

Related Videos

BINOMIAL

See Also

• BINOMDIST
• BINOMIALDIST

References

• Binomial Distribution
• Binomial Coefficient

• List of Main Z Functions

• Z3 home