Ok. So now you are given
The probability mass function of is
However, you can write a more condensed product form:
such that the joint log-likelihood for n sample
which can be readily obtained from your sample.
Now we take the partial differentiation with respect to the parameters
and set them equal to 0:
Simplify them, we have a 2 by 2 linear system in :
Eventually, we have
But note that a, b, c are just counts of the outcomes 1, 2, 3.
So the MLE are just the ratio of the specific counts to the total sample.