0!=1
I could just say that it is define that way which is certainly true.
or I could say that 0! is the number of ways that 0 items can be odered - that would be just one way.
Now
5C1 is the number of ways that 1 item can be chosen from 5 it equals $$^5C_1=\frac{5!}{1!(5-1)!}=\frac{5!}{1*4!}=5\\\\$$
5C0 is the number of ways that 0 item can be chosen from 5 it equals [the answer has to be 1 there is only one way of chosing nothing]
$$\\^5C_0=\frac{5!}{0!(5-0)!}=\frac{5!}{0!*5!}=\frac{1}{0!}\\\\
$ the only way this can work is if 0!=1$\\\\
so\\\\
0!=1$$
Here's an informal "proof"....
3! = 4! / 4
2! = 3! / 3
1! = 2!/ 2
So, by extension
0! = 1! / 1 = 1
0!=1
I could just say that it is define that way which is certainly true.
or I could say that 0! is the number of ways that 0 items can be odered - that would be just one way.
Now
5C1 is the number of ways that 1 item can be chosen from 5 it equals $$^5C_1=\frac{5!}{1!(5-1)!}=\frac{5!}{1*4!}=5\\\\$$
5C0 is the number of ways that 0 item can be chosen from 5 it equals [the answer has to be 1 there is only one way of chosing nothing]
$$\\^5C_0=\frac{5!}{0!(5-0)!}=\frac{5!}{0!*5!}=\frac{1}{0!}\\\\
$ the only way this can work is if 0!=1$\\\\
so\\\\
0!=1$$