Major city tour problem?

Question

A tourist wants to visit 3 or more of the 5 major cities in india : Delhi,Mumbai,Chennai,Kolkata, and Bangalore.
In how many ways can he plan his tour such that Delhi is always included ?(No city is repeated,the sequence of cities is relevant for a tour.)

a)5(4!+3!+2!)
b)5!+4*4!+6*3!
c)5!+4!+3!
d)5C5+4C3+4C2

which is the answer ?

Answer

First.
“the sequence of cities is relevant for a tour”
means order is important. Thus, a permutation reflects each different tour.

There are 3 possibilities for the number of cities included for the tour, namely, 3,4, or 5 cities. We consider each case separately.

Case 1. ( 3 cities )

Delhi is included. The remaining 2 cities can be chosen from the remaining 4 cities in 4C2 ways = 6 ways.

After the 3 cities are selected, they can be permutated in
3P3 ways = 3! ways. Each representing a tour.

Hence, Total number of tours in case 1 = 6 x 3! ways.

Case 2. ( 4 cities )

Delhi is included. The remaining 3 cities can be chosen from the remaining 4 cities in 4C3 ways = 4 ways.

After the 4 cities are selected, they can be permutated in
4P4 ways = 4! ways. Each representing a tour.

Hence, Total number of tours in case 2 = 4 x 4! ways.

Case 3. ( 5 cities )

Delhi is included. The remaining 4 cities can be chosen from the remaining 4 cities in 4C4 ways = 1 way.

After the 5 cities are selected, they can be permutated in
5P5 ways = 5! ways. Each representing a tour.

Hence, Total number of tours in case 3 = 1 x 5! ways.

Thus,

Total number of
“major city tour is”

= “no.tour in case 1,2 and 3.”

= 6 x 3! + 4 x 4! + 5!

= 5!+4*4!+6*3! ways.

The answer is (b)