Question

Sally lives in City A and works for the department of transportation. She needs to inspect the weigh stations in each of four different cities. In how many ways can she reach each of these cities and return to her home in City​ A?

Answer 1

Answer:

She can visit the cities in 192 ways

Step-by-step explanation:

If Sally starts from city A and after going to 4 cities she returns to home,

then that can be in 4! = 24 ways. (permutation of 4 distinct objects)

If between the time she visits cities, she comes home once,

then cities can be visited in,

$4! \times 3_{C_{1}}$ = 72 ways.  (permutation of 4      distinct objects  and choosing 1 gap among the 3 )

If between the time she visits cities, she comes home twice,

then cities can be visited in,

$4! \times 3_{C_{2}}$ =72 ways. (permutation of 4 distinct objects and finding 2 gaps among the 3)

If between the time she visits cities, she comes home thrice,

then cities can be visited in,

$4! \times 3_{C_{3}}$ = 24 ways. (permutation of 4 distinct objects and choosing 3 gaps among the 3)

Now, she can't come home more than thrice in between the time she visits cities (since there are 4 cities to visit only)

So, the number of ways she can visit cities = (24+ 72 + 72 + 24)

                                                                       = 192