what is the probability that a 4-digit license plate will have exactly one pair of numbers if all digits 0-9 are equally likely to appear?

Hi,

sorry I don't quite understand ( due to my English skills .. ) what you mean
with "one pair of numbers".

If you mean all possible combinations of 4-digit numbers that can be formed
from numbers from 0 to 9 are awailable, the answer is 1/10 000. Because
license plate can have numbers 0000, 0001, 0002, ... 1000, 1001, 1002 ...
9999. All combinations makes 10 000 different 4-digit numbers.

If you mean what is the possibility to get 4 numbers "right" when
you use numbers from 0 to 9. Thats only 210 variations of numbers.
This is the case which is similar as lottery. So think that you have only
10 possible numbers and you have to find a row "4-right".
At this case the order of digits doesn't matter ; 0278, 7208, 8270, 8207 ...
are all the same. Each of those have the "right" numbers 0, 2, 7 and 8.
This kind of combinations you can make from 10 digits only 210
different combinations when using 4-digit numbers.
At this case the possibility is 1/210.
You can find more info from Excel-help topic : Combination-function.