Notes: `((100),(98))` Find the binomial coefficient.

Wednesday, December 2, 2015

$\displaystyle{\left(\matrix{{100}\\{98}}\right)}$ 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 should notice that $\displaystyle{\left(\matrix{{n}\\{k}}\right)}={\left(\matrix{{n}\\{n}-{k}}\right)}$

Hence, $\displaystyle{\left(\matrix{{100}\\{2}}\right)}={\left(\matrix{{100}\\{100}-{2}}\right)}={\left(\matrix{{100}\\{98}}\right)}$

Evaluating $\displaystyle{\left(\matrix{{100}\\{2}}\right)}$ yields:

$\displaystyle{\left(\matrix{{100}\\{2}}\right)}=\frac{{{100}!}}{{{2}!{\left({100}-{2}\right)}!}}\Rightarrow{\left(\matrix{{100}\\{2}}\right)}=\frac{{{100}!}}{{{2}!{\left({98}\right)}!}}$

Notice that $\displaystyle{100}!={98}!\cdot{99}\cdot{100}$

$\displaystyle{\left(\matrix{{100}\\{2}}\right)}=\frac{{{98}!\cdot{99}\cdot{100}}}{{{2}!{\left({98}\right)}!}}\Rightarrow{\left(\matrix{{100}\\{2}}\right)}=\frac{{{99}\cdot{100}}}{{{1}\cdot{2}}}={4950}$

Hence, evaluating the binomial coefficient yields $\displaystyle{\left(\matrix{{100}\\{98}}\right)}={4950}.$

Notes

Wednesday, December 2, 2015

$\displaystyle{\left(\matrix{{100}\\{98}}\right)}$ Find the binomial coefficient.

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