The question states:
"Suppose you have a class of 20 students and the professor decides to assign the students into 5 groups to each work on different projects. The group sizes are 3,4,5,4,4. How many possibilities are there for the formation of 5 groups"
My answer:
This is a direct example of the law of ordered partitions:
$\binom{20}{3,4,5,4,4}$ possibilities for groups.
The question then further states that the groups of four are assigned identical projects. So I get this answer:
$\binom{20}{3,4,5,4,4}$$\frac{1}{3!}$
Here is my question/thought. In this case the group numbers are all 4 that are assigned the same project. Would this answer change if the groups who got the identical projects were say the groups of 3,4, and 5? Would you still get $\binom{20}{3,4,5,4,4}$$\frac{1}{3!}$ ?