There are six roads between A and B and 4 roads between B and C.

(i).In how many ways can one drive the circular trip
(ii).Without using the same road more than once.

Hint: In this problem we first find how we can reach from A to B and then from B to C and then back to A via B. For each of the journeys we have to choose one road at a time. Hence, we will use the combination methodology among the mentioned number of roads between any path.

Complete step-by-step answer:

We are given that there are six roads between A and B and 4 roads between B and C. This implies that for the part (i), we will find in how many ways we can reach from A to B, then B to C, then back to B and then back to A.

Number of ways from A to B = ${}^{6}{{C}_{1}}=6$.

Number of ways from B to C = ${}^{4}{{C}_{1}}=4$.

Number of ways from C to B = ${}^{4}{{C}_{1}}=4$.

Number of ways from B to A = ${}^{6}{{C}_{1}}=6$.

Therefore, the total number of ways for a circular trip = $6\times 4\times 4\times 6=576$.

Now, for part (ii), we will calculate in the similar manner but there should be no repetition of roads. Hence the road which is previously used for going to another location cannot be used for returning back.

Number of ways from A to B = ${}^{6}{{C}_{1}}=6$.

Number of ways from B to C = ${}^{4}{{C}_{1}}=4$.

Number of ways from C to B, without using the previously used routes = ${}^{3}{{C}_{1}}=3$.

Number of ways from B to A, without using the previously used routes = ${}^{5}{{C}_{1}}=5$.

Therefore, the total number of ways for a circular trip, without repetition = $6\times 4\times 3\times 5=360$.

Therefore, correct answers are 576 and 360.

Note: The key step in solving this problem is the application of combination with and without restrictions. This implies that for the first part we calculated the total ways possible for a simple circular route. And in the next part, we calculated the route with restriction of repetition. While applying restrictions, the correct condition must be employed to obtain a final answer.