Difference between revisions of "Binomial coefficient"
From specialfunctionswiki
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=Properties= | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> $\displaystyle{n \choose k} = {n \choose {n-k}} = (-1)^k {{k-n-1} \choose k}$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> $\displaystyle{{n+1} \choose k} = {n \choose k} + {n \choose {k-1}}$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> ${n \choose 0} = {n \choose n} = 1$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> $1 + \displaystyle {n \choose 1} + {n \choose 2} + \ldots + {n \choose n} = 2^n$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> $1 - \displaystyle {n \choose 1} + {n \choose 2} - \ldots + (-1)^n {n \choose n} =0$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
<strong>Theorem (Binomial Theorem):</strong> $(a+b)^n = \displaystyle\sum_{k=0}^n {n \choose k} a^k b^{n-k}$ | <strong>Theorem (Binomial Theorem):</strong> $(a+b)^n = \displaystyle\sum_{k=0}^n {n \choose k} a^k b^{n-k}$ | ||
Revision as of 05:27, 11 January 2015
The binomial coefficients are defined by the formula $${}_nC_k:={n \choose k} = \dfrac{n!}{(n-k)!k!}.$$
Properties
Proposition: $\displaystyle{n \choose k} = {n \choose {n-k}} = (-1)^k {{k-n-1} \choose k}$
Proof: █
Proposition: $\displaystyle{{n+1} \choose k} = {n \choose k} + {n \choose {k-1}}$
Proof: █
Proposition: ${n \choose 0} = {n \choose n} = 1$
Proof: █
Proposition: $1 + \displaystyle {n \choose 1} + {n \choose 2} + \ldots + {n \choose n} = 2^n$
Proof: █
Proposition: $1 - \displaystyle {n \choose 1} + {n \choose 2} - \ldots + (-1)^n {n \choose n} =0$
Proof: █
Theorem (Binomial Theorem): $(a+b)^n = \displaystyle\sum_{k=0}^n {n \choose k} a^k b^{n-k}$
Proof: █
