Calculate Permutations of Any Number of Letters in a Name - Explained

[samiiv]

Homework Statement

Michelle knows that there are 8!/2!2! permutations of her name when ALL the letters are used.
She would like to know how many permutations there are if ANY NUMBER OF LETTERS in her name are used. Explain your procedure.

The Attempt at a Solution

this is what i figured out, i don't think I'm right though.

if your using all 8 letters there are 8! ways, but if only 7 letters are used than there are 8x7x6x5x4x3x2 ways,and if 6 letters are used than there are 8x7x6x5x4x3 and if 5 letters 8x7x6x5x4 ways all the way down to 1 letter. than the product of each are added since they are mutually exclusive.

 

[ArcanaNoir]

It's correct to add all those like you do, but notice you start with 8! When you are told in the problem that using all of the letters in her name is 8!/2!2!. So you need to remember to adjust for the fact that some letters appear twice and something like el and el should only count once, not twice, even though it will happen twice since there are two e's and two l's.