Question 1210184
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I can only remember a seven-digit telephone number if the first three digits (the "prefix") 
are equal to the next three digits or the last three digits. 
For example, I can remember 389-3892 and 274-9274.

How many seven-digit telephone numbers can I remember?

(For this problem, a telephone number cannot start with a 0.)
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            Counting in the post by @CPhill is  INCORRECT.



It becomes incorrect, when CPhill starts consider the overlapping case.
Namely, he suddenly considers then the 9-digit telephone numbers instead of 7-digit numbers.


In reality, the overlapping case may really have place.


It happens when in the form  ABC-XABC part XAB is the same as ABC.

But it happens if and only if  A=X, B=A, C = B, i.e.  A = B = C.

The number of such cases (overlapping) is 9  (not 900, as @CPhill counts them).


Therefore, the final answer in the problem is  9000 + 9000 - 9 = 18000-9 = 17991.


<U>ANSWER</U>.  You can remember 17991 telephone numbers.


Solved correctly.