Consider a system of three harmonic oscillators (shos), each with possible energies 0, 1ε, 2ε, 3ε, 1) what is the number of microstates for total energy u = 3ε ? number =

To find the number of microstates means to find in how many ways you can distribute M quanta of energy in N oscillators. This is given by a binomial distribution:
         M + N - 1
Ω =  (               ) =  
               M

In your case: N = 3 and M = 3, therefore:

Ω = 5! / (2! 3!) = 10

Ais the answer for your question