Question

PLEASE HELP!!

My friend lost 2 charms off her 7-charm bracelet. For her birthday, I bought her a new charm to replace one of the lost ones. Unfortunately, I messed up and got her a duplicate of one of the charms she still has. How many distinguishable ways can she put her 6 charms on her bracelet (remember that two of the charms are the same)?

Answer 1

So you have 6 charms, call them abcdef.
Except that a = f, so you really have abcdea

If we had abcdef, then there would be 6! possible orderings.
But every ordering like say,
abcfde is the same as fbcade

so you have to divide by 2.
At least, that's what I think.

Answer 2

actually since there are 6 charms all in all you have 6! then one of the charms repeat twice so divide by 2! giving you \[\frac{6!}{2!}\]