Answer:
112
Explanation:
You want the number of 3-digit "dropdown numbers".
Dropdown number
A "dropdown number" has distinct digits, one of which is the average of all digits.
We can count the number of dropdown numbers by considering the digit that is the average of the rest. For that digit (n), the number of possible choices of the other digits is the lesser of (n) or (9-n). For example, for n=2, we can have other digit pairs of (0, 4) or (1, 3). For n=6, we can have other digit pairs of (5, 7) or (4, 8) or (3, 9).
Counting
The value of n may range from 1–8, inclusive, so the number of digit triples will be 2(1 +2 +3 +4) = 20. Each triple of digits can be arranged 6 ways, for a total of 20·6 = 120 numbers of 3-digits.
Of those, 8 numbers will have a leading 0, so may not count as 3-digit numbers. That leave a total of 120 -8 = 112 3-digit dropdown numbers.
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Additional comment
The first 5 in numerical order are ...
102, 120, 123, 132, 135
The last 5 are ...
957, 963, 975, 978, 987
In general, a number on the list, m, has a corresponding number 999-m also on the list. The exceptions are numbers that have a leading digit of 0.