What are all of the possible outcomes that satisfy that condition?

# of 1's # of 4's # of (not 1's, not 4's) 3 0 4 4 1 2 5 2 0

Those are the only outcomes that you care about. So, what are the probabilities of each?

I will help you out for the first case, see if you can figure out the other two. The probability that three out of the seven dice will roll a 1 and the other four will roll neither a 1 nor a 4: