Difference between revisions of "Binomial coefficient"

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=Properties=
 
=Properties=
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<strong>Proposition:</strong> $\displaystyle{n \choose k} = {n \choose {n-k}} = (-1)^k {{k-n-1} \choose k}$
 
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<strong>Proof:</strong> █
 
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<strong>Proposition:</strong> $\displaystyle{{n+1} \choose k} = {n \choose k} + {n \choose {k-1}}$
 
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<strong>Proof:</strong> █
 
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<strong>Proposition:</strong> ${n \choose 0} = {n \choose n} = 1$
 
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<strong>Proof:</strong> █
 
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<strong>Proposition:</strong> $1 + \displaystyle {n \choose 1} + {n \choose 2} + \ldots + {n \choose n} = 2^n$
 
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<strong>Proof:</strong> █
 
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<strong>Proposition:</strong> $1 - \displaystyle {n \choose 1}  + {n \choose 2} - \ldots + (-1)^n {n \choose n} =0$
 
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<strong>Proof:</strong> █
 
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[[Binomial theorem]]
 
[[Binomial theorem]]
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[[Binomial coefficient (n choose k) equals (n choose (n-k))]]<br />
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[[Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)]]<br />
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[[Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1))]]<br />
 +
[[Binomial coefficient (n choose 0) equals 1]]<br />
 +
[[Binomial coefficient (n choose n) equals 1]]<br />
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[[Sum over bottom of binomial coefficient with top fixed equals 2^n]]<br />
 +
[[Alternating sum over bottom of binomial coefficient with top fixed equals 0]]<br />
  
 
=Videos=
 
=Videos=

Revision as of 02:57, 4 June 2016

The binomial coefficients are defined by the formula $${}_nC_k:={n \choose k} = \dfrac{n!}{(n-k)!k!}.$$


Properties

Binomial theorem Binomial coefficient (n choose k) equals (n choose (n-k))
Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)
Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1))
Binomial coefficient (n choose 0) equals 1
Binomial coefficient (n choose n) equals 1
Sum over bottom of binomial coefficient with top fixed equals 2^n
Alternating sum over bottom of binomial coefficient with top fixed equals 0

Videos

Pascal's Triangle and the Binomial Coefficients
Example of choose function (Binomial Coefficient)
Binomial coefficients

References