Double factorial
I was scrolling through old water cooler topics looking for something I vaguely remember from years ago and didn’t find what I was looking for but did come across some old ``Math Problems’’ from 4 or 5 years ago where there were running jokes about knowing when ! means factorial and when it is just punctuation.
At some point !! appeared. I suspect a few people don’t realize that (as the link puts it) the double factorial n!! is not the same as applying the factorial twice (n!)!
n!=n\times(n-1)\times\cdots\times2\times1 so that 5!=5\times4\times3\times2\times1=120.
For even numbers, (2n)!!=(2n)\times(2n-2)\times\cdots\times4\times2 so that 6!!=6\times 4\times 2=48.
For odd numbers, (2n-1)!!=(2n-1)\times(2n-3)\times\cdots\times3\times1 so that 5!!=5\times 3\times 1=15.
That is all.