Posted by Randy on Wednesday, May 23, 2012 at 5:53pm.
So the template is
NXXNNNN
I interpret the "repeated digit" as any two identical digits appearing anywhere on the licence plate, as opposed to two identical digits next to each other.
Also, it was not specified as "exactly one repeated letter or digit", so we assume multiple repetitions are acceptable, such as 5AA5002, or 4BB3344, etc.
First, we find the size of the sample space (Ω)
N=|Ω| = 10*26*26*10*10*10*10
=67,600,000
It is easier to find the number of licence plates that have distinct digits and letters, namely:
n=10*26*25*9*8*7*6=19,656,000
Therefore, number of licence plates with at least one repeated digit or letters is
67,600,000-19,656,000
=47,944,000