There are four roads from city X to Y and five roads from city Y to Z, find (i) how many ways is it possible to travel from city X to city Z via city Y. (ii) how different round trip routes are there from city X to Y to Z to Y and back to X.

(i)

Number of ways from $X$ to $Y$ is $4.$

Number of ways from $Y$ to $Z$ is $5.$

Then number of ways from $X$ to $Z$ passing $Y$ is $4\cdot 5=20.$

(ii)

Number of ways from $X$ to $Z$ passing $Y$ is $4\cdot 5=20.$

Number of ways from $Z$ to $Y$ without using the same road more than once is $5-1=4.$

Number of ways from $Y$ to $X$ without using the same road more than once is $4-1=3.$

Number of ways from $Z$ to $X$ passing $Y$ without using the same road more than once is $4\cdot 3=12.$

Number of different round trip routes are there from city $X$ to $Z$ passing $Y$ and back from city $Z$ to $X$ passing $Y$ is

$20\cdot12=240.$