xc630 said:
Hello, I having toruble iwth this following problem. It says: How does the formula for nCn suggest the definition 0!=1?
When I plug in I get n!/(n-n)!n! but this doe not lead me anywhere. I would appreciate any help.
I assume you started with the formula for <sub>n</sub>C<sub>r</sub> (the number of combinations of n things taken r at a time):
<sub>n</sub>C<sub>r</sub> = n! / [r! * (n - r)!]
So, <sub>n</sub>C<sub>n</sub> would be (substituting "n" for "r"):
<sub>n</sub>C<sub>n</sub> = n! / [n! * (n - n)!]
or,
<sub>n</sub>C<sub>n</sub> = n! / [n! * 0!]
You can divide numerator and denominator by n!, leaving us with this:
<sub>n</sub>C<sub>n</sub> = 1 / 0!
But we know that there is exactly
1 way to form a group of n items from n items; in other words, that <sub>n</sub>C<sub>n</sub> = 1. So,
1 = 1 / 0!
The
only way that this can be true is for 0! to be 1.....thus the definition that 0! = 1.
I hope this helps you.
