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September 23, 2014

Posted by **marie** on Monday, August 22, 2011 at 12:41pm.

106!

104!

- math -
**MathMate**, Monday, August 22, 2011 at 4:18pmThere is no

*exact*way to evaluate factorials except to calculate them by multiplication:

They turn out to be:

104!=102990167451456276238485838647650442830537724549990721823254917768878717324

75287174542709871683888003235965704141638377695179741979175588724736000000000000

000000000000

=1.0299016745145628*10^166 (approx.)

and

106!=114628056373470835453434738414834942870388487424139673389282723476762012382

44994625266036087184167347601629828709643514374735052822822430250631168000000000

0000000000000000

=1.1462805637347084*10^170 (approx.)

If only an approximation is required, Stirling's approximation could be used for large factorials (n>100):

n! ~ sqrt(2πn)(n/e)^n

where the error is of the order 1/n.

A closer approximation can be obtained by summing the series:

n! ~ sqrt(2πn)(n/e)^n(1+1/(12n)+1/(288n²)-139/(51840n³)-...)

For more information, see:

http://en.wikipedia.org/wiki/Stirling%27s_approximation

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