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Digitally Average Integers (Posted on 2019-08-14) Difficulty: 3 of 5
We will say that an integer is digitally average if the average of its (decimal) digits is 4.5 (the average of the digits from 0 to 9). How many 10-digit integers are digitally average?

No Solution Yet Submitted by Danish Ahmed Khan    
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solution | Comment 1 of 2
we can use generating functions to solve this problem.  If we allowed leading zeros then we could simply use the coefficient of x^45 from the expansion of (1+x+x^2+...+x^9)^10 which turns out to be 432457640.  To eliminate those which have leading zeros we can subtract the coefficient of x^45 from (1+x+x^2+...+x^9)^9 which is 40051495.

Subtracting these two we get

so there are 392,406,145 such 10-digit numbers

  Posted by Daniel on 2019-08-14 08:41:31
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