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 10digit integers are digitally average?
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
43245764040051495=392406145
so there are 392,406,145 such 10digit numbers

Posted by Daniel
on 20190814 08:41:31 