Notes: `((10),(6))` Find the binomial coefficient.

Thursday, April 2, 2009

$\displaystyle{\left(\matrix{{10}\\{6}}\right)}$ Find the binomial coefficient.

You need to evaluate the binomial coefficient, using the next formula, such that:

$\displaystyle{\left(\matrix{{n}\\{k}}\right)}=\frac{{{n}!}}{{{k}!{\left({n}-{k}\right)}!}}$

You need to replace 10 for n and 6 for k, such that:

$\displaystyle{\left(\matrix{{10}\\{6}}\right)}=\frac{{{10}!}}{{{6}!{\left({10}-{6}!\right)}}}\Rightarrow{\left(\matrix{{10}\\{6}}\right)}=\frac{{{10}!}}{{{6}!{4}!}}$

Notice that $\displaystyle{10}!={6}!\cdot{7}\cdot{8}\cdot{9}\cdot{10}$

$\displaystyle\frac{{{10}!}}{{{6}!{4}!}}=\frac{{{6}!\cdot{7}\cdot{8}\cdot{9}\cdot{10}}}{{{6}!{4}!}}\Rightarrow{\left(\matrix{{10}\\{6}}\right)}=\frac{{{7}\cdot{8}\cdot{9}\cdot{10}}}{{{1}\cdot{2}\cdot{3}\cdot{4}}}={210}$

Hence, evaluating the binomial coefficient yields $\displaystyle{\left(\matrix{{10}\\{6}}\right)}={210}.$

Notes

Thursday, April 2, 2009

$\displaystyle{\left(\matrix{{10}\\{6}}\right)}$ Find the binomial coefficient.

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