Could someone help decode what this combinatoric problem is asking me? The problem:

There are $10$ professors at a certain CS department. According the tentative course schedule, there are $7$ distinct courses that should be taught next semester. Please count in how many ways the teaching assignments can be distributed among the department faculty if:
  (a) each professor should teach at most one class?
  (b) all $7$ classes must be taught by either Professor Lamport or Professor Papadimitriou.
  (Clarification: In both cases, you cannot assign two professors to teach the same class together)

I'm actually not entirely sure what I'm being asked to do. I asked a classmate and he said (a) was asking us to distribute the $7$ classes across ten possible Professors, which I think would be $10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4$ ways. To be honest, I'm not sure how my classmate drew that conclusion (am I reading these incorrectly?). As for the second question, I again am not sure exactly what I'm being asked to consider. Is it as simple as saying there are two professors and each of them can take $3$ to $4$ of the $7$ courses, and I'm supposed to compute the number of possible permutations? 
 A: For part (a) the answer is indeed
$$
_{10}P_7 = 10\cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 = 201\,600
$$
The reason is this: You have ten professors to chose from to lead course number $1$. No matter which professor you choose, there are $9$ professors left to chose from when it comes to course two. That makes $10\cdot 9$ total different ways of assigning professors to course one and two. Then comes course three, with $8$ options left. And so on.
The second question says that only Lampart or Papadimitriou are available to teach, but doesn't have any other limitations (don't assume that there is a $3$-to-$4$ distribution unless you're told that there is). That means that Lampart could take all seven. Or, Papadimitriou could take one, and Lampart six, which is a distribution that can be done in seven different ways (either Papadimitriou has course one, or he has course two, or...). And so on.
There is one way of counting all these combinations for part (b) that is easier than the one I've outlined above. Again, we focus on the classes instead. The first course can be taught by either Lampart of Papadimitriou. That's two possibilities. The second course can also be taught by Lampart or Papadimitriou, which makes two possibilities for course two, regardless of what happened to course one. That means that there is a total of $2\cdot 2$ ways of setting up the lecturers for the first two courses. Then comes course three, with two more possibilities, and so on. In the end, the answer is
$$
2^{7} = 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2 = 128
$$
