Notes: `_8 C_6` Find the binomial coefficient.

Saturday, August 9, 2008

$\displaystyle_{8}{C}_{{6}}$ Find the binomial coefficient.

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

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

Replacing 8 for n and 6 for k yields:

$\displaystyle{8}{C}{6}=\frac{{{8}!}}{{{6}!{\left({8}-{6}\right)}!}}\Rightarrow{8}{C}{6}=\frac{{{8}!}}{{{6}!\cdot{2}!}}$

Notice that $\displaystyle{8}!={1}\cdot{2}\cdot{3}\cdot{4}\cdot{5}\cdot{6}\cdot{7}\cdot{8}\Rightarrow{8}!={6}!\cdot{7}\cdot{8}$

$\displaystyle{8}{C}{6}=\frac{{{6}!\cdot{7}\cdot{8}}}{{{6}!\cdot{2}!}}\Rightarrow{8}{C}{6}=\frac{{{7}\cdot{8}}}{{{2}!}}\Rightarrow{8}{C}{6}=\frac{{{7}\cdot{8}}}{{{1}\cdot{2}}}$

Hence, evaluating the binomial coefficient yields 8C6 = 28.

Notes

Saturday, August 9, 2008

$\displaystyle_{8}{C}_{{6}}$ Find the binomial coefficient.

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