Circular permutations problem, need help 
How many ways can $5$ people sit on a circular table of $8$ places?

I know that if both people and  places number =$n$ , then the answer $= (n-1)!$
But the situation here is different, since there are more places than people,  what should I do?  
Thanks for help
 A: Fix the position of person $1$. Then there are $7$ remaining options for person $2$. Next, there are $6$ options for person $3$, and continue down. The idea is similar to the proof for $(n-1)!$ you mention in the OP.
A: There’s more than one way to interpret the question, as stated, but the answers are related. There are $\binom85=56$ different choices of $5$ seats, so if it’s just about which places are taken, that’s your answer.
If rotations of the same arrangement are to be considered equivalent (else, why specify a round table?), then we divide by $8$, and there are only $7$ really different-looking ways to choose $5$ from an $8$-circle. (One with all $5$ adjacent, two with a group of $4$ and a singleton, three with a triplet, and one with no groups $>2$.)
If the people in the seats are distinguishable, then we multiply one of the above answers by $5!$.
If I were writing this problem, I’d think of the version where rotations are indistinguishable but people are not is the most “interesting”, or “natural”, or in some way appealing to me. That in no way guarantees that it is what you are being asked.
A: Let $X$ be the set of all possible arrangements of $5$ people in a straight line of $8$ chairs. Now, define a relation on $X$ : two arrangements are similar, if they are related by a cyclic permutation: if there exists a number $0 \leq r\leq 7 $, such that each person in the first arrangement, when shifted by $r$ places to the right, lands up where he is supposed to be in the second arrangement (with wrapping around i.e. from extreme right the next shift is to the extreme left).
Now, note that this relation is an equivalence relation, and the equivalence class of each straight line permutation contains eight elements, and represents one single arrangement around a circular table.
Therefore, the answer to your question is just the number of ways that $5$ people can sit in $8$ places in a straight line(that is, with no constraints on symmetry), divided by $8$. The answer to this question is then just $\frac{8 \times7\times6\times 5\times 4}{8} = 7\times6\times5\times4 = 840$.
