Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 2"
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*[[Mock AIME 1 2006-2007]] | *[[Mock AIME 1 2006-2007]] | ||
[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
Latest revision as of 14:53, 3 April 2012
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 
.
Solution
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 
.
