The number of arrangements in which Veena can visit the 4 cities A, B, C, D

                           = 

The sample space of the 24 arrangements is:

                       S = 

                 n(S) = 24

(i) Event E: She visits A before B.
                             E = {ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, CABD, CADB, CDAB, DABC, DACB, DCAB}
                        n(E) = 12
Hence,                   

(ii) Event F: Veena visits A before B and B before C.
                       F = {ABCD, ABDC, ADBC, DABC}
                     n(F) = 4
Hence,       

(iii) Event G: Veena Visits A first and B last
                           G = {ACDB, ADCB}
                        n(G) = 2
Hence,                  

(iv) Event H: Veena visits A either first or second.
                              H = {ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, CABD, CADB, DACB, DABC}
                            n(H) = 12
Hence,                    

(v) Event k: Veena visits A just before B.
                K = {ABCD, ABDC, CABD, CDAB, DABC, DCAB}
             n(K) = 6
Hence,