# Number of different license plates

1. Apr 25, 2010

### IniquiTrance

My book says that if a license plate must have 3 letters, and 4 numbers, when repetition is allowed, and no restriction is placed on the order of letters and numbers, then the amount of possible license plates is:

$$\begin{pmatrix}7 \\3\end{pmatrix}10^{4}26^{3}$$

$$\begin{pmatrix}7 \\3\end{pmatrix}$$ ways of permuting the letters and numbers, seems reasonable when all the letters and numbers are the same, such as:

AAA1111

But shouldn't there be 7! possible unique permutations when they're all different, such as in say:

ABC1234

Thanks!

Last edited: Apr 25, 2010
2. Apr 25, 2010

### tiny-tim

Hi IniquiTrance!

(try using the X2 tag just above the Reply box )
No …

104263 is the number of ways of choosing a three-letter word and a four-figure number …

the order matters (so ABC is different from CBA), but then you have to decide which three places to put the three letters, so that's the number of ways of choosing 3 out of 7, which is 7C3.

Alternatively, start with the choice of the three places … that's 7C3, and the order doesn't matter, and then you have to decide which word to put in those three places (and which number to put in the other four places), and now the order matters.

(I assume that's what's confusing you is that you're thinking "in 7C3, the order doesn't matter, but in this case the order obviously does matter" … but the order only matters once … if CAT is different from TAC, then counting CAT in positions 123 and 321 would duplicate the counting of TAC. )

3. Apr 25, 2010

### IniquiTrance

Thanks Tim! Great explanation, as per usual.