Notes: `_5 C_3` Find the binomial coefficient.

Monday, June 18, 2012

$\displaystyle_{5}{C}_{{3}}$ 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 5 for n and 3 for k yields:

$\displaystyle{5}{C}{3}=\frac{{{5}!}}{{{3}!{\left({5}-{3}\right)}!}}\Rightarrow{5}{C}{3}=\frac{{{5}!}}{{{3}!\cdot{2}!}}$

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

$\displaystyle{5}{C}{3}=\frac{{{3}!\cdot{4}\cdot{5}}}{{{3}!\cdot{2}!}}\Rightarrow{5}{C}{3}=\frac{{{4}\cdot{5}}}{{{2}!}}\Rightarrow{5}{C}{3}=\frac{{{4}\cdot{5}}}{{{1}\cdot{2}}}={10}$

Hence, evaluating the binomial coefficient yields $\displaystyle{5}{C}{3}={10}.$

Notes

Monday, June 18, 2012

$\displaystyle_{5}{C}_{{3}}$ Find the binomial coefficient.

No comments:

Post a Comment

How does the choice of details set the tone of the sermon?