3. How many triples $(A, B, C)$ of positive integers (positive integers are the numbers $1,2,3,4, \ldots$ ) are there such that $A+B+C=10$, where order does not matter (for instance the triples $(2,3,5)$ and $(3,2,5)$ are considered to be the same triple) and where two of the integers in a triple could be the same (for instance $(3,3,4)$ is a valid triple).
Solution: The triples can merely be enumerated: $(1,1,8),(1,2,7),(1,3,6),(1,4,5)$, $(2,2,6),(2,3,5),(2,4,4)$, and $(3,3,4)$. There are 8 elements.