Number of different license plates

[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!

 

[tiny-tim]

Hi IniquiTrance!

(try using the X² tag just above the Reply box )

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

ABC1234

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. )

 

[IniquiTrance]

Thanks Tim! Great explanation, as per usual.