I’m afraid that you’ll have to start by redoing (a) and (b): both are wrong. Call the friends $A,B$, and $C$.
(a) I’m assuming that what’s wanted here is the number of ways in which they completely miss one another: no two go to the same restaurant. There are $10$ restaurants that $A$ could choose. $B$ could then choose any of the remaining $9$, and $C$ could choose any of the $8$ not chosen by $A$ or $B$. Thus, there are $10\cdot9\cdot8=720$ different ways in which they could completely miss one another.
(b) In order for them to meet successfully, they must all choose the same restaurant. There are $10$ restaurants, so there are just $10$ ways in which they can meet successfully. Alternatively, we can apply the same kind of analysis as in (a): $A$ has a choice of $10$ restaurants, but after that $B$ and $C$ have only $1$ choice each, if the meeting is to take place, so the total number of ways is $10\cdot1\cdot1=10$.
(c) This is most easily done by considering the complementary (opposite) event: no two of them meet. In (a) we calculated the number of ways in which this can happen. If there are $n$ different ways in which could they choose restaurants, regardless of how many of them meet, the number of ways in which at least two of them meet must be $n-720$. What’s $n$?