Counting and Probability why is this wrong?

[charmedbeauty]

Counting and Probability...why is this wrong??

Homework Statement

How many ten letter words can be made from the english alphabet which contain at least one X, at least one Y and at least one Z?

Homework Equations

The Attempt at a Solution

1) Choose a place for X (10 ways)
2)Choose a place for Y(9 ways)
3)Choose a place for Z(8 ways)
4)Choose another seven letters from the alphabet (26) with repetitions allowed (26⁷ ways)

So the number of 10 letter words= 10.9.8.26⁷ where (.) means multiply.

what is so wrong about this?

 

[Infinitum]

Think about this,

One of your words would have be XXYZCDEFGHI, the bolded X being the one you chose initially. Now, you have another word XXYZCDEFGHI, the bolded again being the X you chose this time. Both are the same words. Apply the same logic to other letters, so many cases in your answer are repeated.

 

[Ray Vickson]

Homework Statement

How many ten letter words can be made from the english alphabet which contain at least one X, at least one Y and at least one Z?

Homework Equations

The Attempt at a Solution

1) Choose a place for X (10 ways)
2)Choose a place for Y(9 ways)
3)Choose a place for Z(8 ways)
4)Choose another seven letters from the alphabet (26) with repetitions allowed (26⁷ ways)

So the number of 10 letter words= 10.9.8.26⁷ where (.) means multiply.

what is so wrong about this?

You double-count the words that have two identical letters, or triple-count the ones with three identical letters, etc.

RGV

 

[charmedbeauty]

You double-count the words that have two identical letters, or triple-count the ones with three identical letters, etc.

RGV

Ohh now it seems so clear... is it normal to struggle at counting?