Assuming the licences are of the form $\text{LLLLNNNN}$ where $\text{L}$ and $\text{N}$ denote letters and numbers respectively:
Number of licences with no letters or numbers repeated:
$26\cdot 25\cdot 24\cdot 23\cdot 10\cdot 9\cdot 8\cdot 7$.
So, number of licences with at least one letter or number repeated:
$\text{Total number of licences} - 26\cdot 25\cdot 24\cdot 23\cdot 10\cdot 9\cdot 8\cdot 7$.
Where the total number of licences is $26^4\cdot 10^4$ which is what you calculated.
Assuming the letters and numbers can appear in any order:
You can obtain this answer by multiplying the answer obtained above by $\dfrac{8!}{4!4!}$ which is the number of permutations of $8$ things in which $4$ objects are of one kind and $4$ objects are of the other kind. This is the same as the answer given by Omnitic.
Please note that as pointed out by Omnitic, the following does not restrict the number of letters and digits to $4$:
Assuming the numbers or letters can be filled in any order (and are not restricted to be $4$):
The licences have $8$ places which can be filled in with a number or a letter. Thus each place can have one of $26+10=36$ values.
Now, number of licences with no letters or numbers repeated: $36\cdot 35\cdot 34\cdot\cdot\cdot 30\cdot29$.
So, number of licences with at least one letter or number repeated:
$\text{Total number of licences} - 36\cdot 35\cdot 34\cdot\cdot\cdot 30\cdot29$
The total number is given by $36^8$.
We have a license plate format:
- first digit from $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}:\;$ 9 choices
- second, third, fourth digit from $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}:\;$
10 choices, for each place.
- fifth position: one of 24 letters (26 letters of alphabet, minus the two distinct letters not allowed) gives 24 choices.
Using the rule of the product, that gives us:
$$9 \times 10 \times 10 \times 10 \times 24 = 9 \times 10^3 \times 24 = 216,000\;\text{license plates available}$$
Best Answer
L-L-D-L-D:
You have:
$\quad$ ___options for the first letter
$\times $ ___options for the second letter (not necessarily distinct from the first letter)
$\times$ ___options for the first digit
$\times$ ___options for the last letter, (not necessarily distinct from the first or second letter)
$\times$ ___options for the last digit...(not necessarily distinct from the first digit)
= total number of possible license plates that can be produced in Missouri.