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Factorials!

What about reducing this guy?

73! / 70!

I don't know about you, but I don't really want to write these guys out.  I'm not THAT sick!

Did you notice something cool that happened in our last two examples?

Check it out:

8! / 5! = ( 8 * 7 * 6 * 5! ) / 5! ! ... the 5! cancels out ... = 8 * 7 * 6 = 336


 

6! / 4! = ( 6 * 5 * 4! ) / 4! ... the 4! cancels out ... = 6 * 5 = 30

So...

73! / 70! = ( 73 * 72 * 71 * 70! ) / 70! ... the 70! cancels out ... = 73 * 72 * 71 = 373,176

 

Let's get a little messier:

Let's reduce

5! / ( 2! 3! )

Expand them out:

5! / ( 2! 3! ) = ( 5 * 4 * 3 * 2 * 1 ) / ( 2 * 1 * 3 * 2 * 1 ) ... the 3 * 2 * 1 cancels out ... = ( 5 * 4 ) / ( 2 * 1 ) = 20 / 1 = 10

 

We could have been a little more efficient:
 

5! / ( 2! 3! ) = ( 5 * 4 * 3! ) / ( 2! 3! ) ... the larger of the factorials (the 3!) cancels out ... 20 / 2 = 10

 

TRY IT:

Reduce

10! / ( 2! 8! )


Reduce

100! / ( 3! 97! )