How Many Sets of 5 Can You Make From 22 Elements?

I know you can just say $\large\frac{22!}{(22!-5!)5!}$. My question is deeper than this, however. What if the set of 5 cannot be used again?

For instance, I have 22 coloured balls, my first set of 5 balls is as follows:-

Blue Red Green Yellow White

Also, I can never have a different combination of the same 5 balls, for instance the following could not be counted either:-

Red Blue Green Yellow White

I can't use the same set again, but I can replace one of the elements, like so:-

Blue Red Green Purple White

How many different combinations of 5 can I make from my 22 balls? Is there a formula for this?

1 Answer

This is just $\dbinom{22}{5} = \dfrac{22!}{5!(22-5)!}$. It appears you have gotten the formula wrong. Also, the above formula does not count repeats as you are suggesting it does.