There are seven persons P, Q, R, S, T, U and V are going to the different cities i.e. Delhi, Patna, Bangalore, Ranchi, Goa, Mumbai and Chennai, but not necessarily in the same order and all of them have different designations viz, Engineer, Artist, Doctor, Lawyer, Dentist, Actor and Businessman, but not necessarily in the same order.
The one who is going to Patna is not an Actor and Businessman. Engineer is going to either Bangalore or Chennai.
Neither V nor T is going to Chennai and Patna. The person who is Artist is going to Goa.
R is a Doctor and S is Businessman. Neither T nor P is going to Goa. Dentist is going to Delhi but he is not P.
U is Actor and he is not going to Bangalore and Ranchi. V is going to Ranchi and he is not Engineer.
P is not going to Chennai. Businessman is not going to Mumbai.

Explanation

Step I: Let us start with the detail solution. R is a Doctor and S is Businessman. U is Actor and he is not going to Bangalore and Ranchi. V is going to Ranchi and he is not Engineer.

Step II: The one who is going to Patna is not an Actor and Businessman. Neither T nor P is going to Goa.

Step III: The person who is Artist is going to Goa. So, Q is going to Goa and he is an Artist. Dentist is going to Delhi but he is not P. So, T is a dentist and going to Delhi.

Step IV: Now, remaining P is an engineer and V is Lawyer. Engineer is going to either Bangalore or Chennai. P is not going to Chennai. So, it is clear that P is going to Bangalore. Businessman is not going to Mumbai.
Step V: Now, only remaining R is going to Patna. U is going to Mumbai and S is going to Chennai. So, the final arrangement is-