Difference between revisions of "Manuals/calci/BINOMIALCOEFFICIENT"

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==BInomial==
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<div style="font-size:30px">'''BINOMIAL(n,k)'''</div><br/>
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*<math>n</math>  is the number of items.
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*<math>k </math> is the  number of selection.
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 +
 
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==Description==
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*This function gives the coefficent of the binomial distribution.
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*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.
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*The coefficients satisfy the Pascals recurrence.
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*The binomial coefficents are denoted by <math>\binom{n}{k}</math> and it is read by n choose k.
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*It is the coefficient of the <math>x^k</math> term in the polynomial expansion of the binomial thorem <math>(1 + x)^n</math>.
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*The coefficient is occur in the formula of binomial thorem:
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<math>(x+y)^n=\sum _{k=0}^n \binom{n}{k} x^{n-k} y^k</math> where <math> k\le n</math>.
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*To find the coefficient of the binomial ,we can use several methods.
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  1. Recursive formula
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  2. Multiplicative formula
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  3. Factorial formula.
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*1.Recursive Formula:
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<math>\binom{n}{k}= \binom{n-1}{k-1} +\binom{n-1}{k}</math>  for <math>n,k>0</math> and <math>1\le k\le n-1</math>.
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*2. Multiplicative formula:
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<math>\binom{n}{k}= \prod_{i=1}^k  \frac{n+1-i}{i}</math>
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*3.Factorial formula:
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<math>\binom{n}{k}= \frac{n!}{k!(n-k)!}</math> where <math>k\le n</math>,and which is zero when <math>k>n</math>.
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*Also  for the initial values <math> \binom{n}{0}=\binom{n}{n}=1 </math> for <math>n\ge 0</math>.
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*Most compact  formula for the coefficient of the binomial value is Factorial formula.
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*Factorial formula is symmetric of the combination formula.
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==ZOS==
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*The syntax is to calculate BINOMIAL in ZOS is <math>BINOMIAL (a,b)</math>.
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**<math>a</math>  is the number of items.
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**<math>b</math> is the  number of selection.
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*For e.g., BINOMIAL(20..25,4)
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*BINOMIAL(10..14,7..8)
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==Examples==
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#=BINOMIAL(10,3)= 120
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#=BINOMIAL(32,0)= 1
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#=BINOMIAL(10,7) = 120
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==See Also==
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*[[Manuals/calci/BINOMDIST  | BINOMDIST ]]
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*[[Manuals/calci/BINOMDIST  | BINOMIALDIST ]]
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==References==
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*[http://en.wikipedia.org/wiki/Binomial_distribution Binomial Distribution]
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*[http://en.wikipedia.org/wiki/Binomial_coefficient Binomial Coefficient]

Revision as of 15:29, 21 November 2016

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 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^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 .
  • 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 (a,b)} .
    • 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 a} 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 b} 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(32,0)= 1
  3. =BINOMIAL(10,7) = 120

See Also

References

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