Why is the probability multiplied by $\binom{n}{k}$ A while ago I asked a question about probability here Why is binomial probability used here?
I get that you can find how many ways of choosing the $6$ correct out of $10$ questions.
But why do we multiply by $\binom{n}{k}$? 
I thought it is simple casework that:
$$P(\text{total probability}) = P(\text{Q 1->6 right and 7-10 wrong} + P(\text{Q 1->5 right, 6 wrong, 7 right, 8-10 wrong}) + ...$$
What is the idea? 

You are correct that the simple casework will work. However, if you were to look carefully at the casework, you will notice something interesting. In each case, the actual probabilities are the same—you will have 6 correct and 4 incorrect answers. The only thing that changes is which of the 10 are correct or wrong.
So, you are looking for all of the patterns using just the letters "C" and "I" (for example) of length 10 where 6 are "C" and 4 are "I". Your cases becomes adding CCCCCCIIII and CCCCCIIIIC and CCCCIIIICC etc.
Guess what, the number of cases you will have to add individually are the number of ways to order 6 success in 10 events, or ${10 \choose 6}$. So all the multiplication is doing is saving you a whole bunch of additions. Well, that is really all what multiplication is—repeated application of addition. Since we know how many identical items we have to add, we multiply.
