Mock AIME 1 2006-2007 Problems/Problem 2
Let be the sum of the digits of a positive integer . is the set of positive integers such that for all elements in , we have that and . If is the number of elements in , compute .
Equivalently, we need to place 12 indistinguishable balls into 7 distinguishable boxes so that no box contains more than 9 balls. There are ways to place 12 objects into 7 boxes. Of these, 7 place all 12 into a single box. place 11 in one box and 1 in a second. place 10 into one box and 2 into a second. place 10 into one box and 1 into each of two others. Thus, this gives us so .