Question

A sports complex has an entrance, indoor stadium, gym and swimming pool with the paths connecting them as shown above. In how many ways can a person go from entrance to swimming pool?

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Answer 1

Let E represents Entrance

I represents indoor stadium

G represents Gym

S represents Swimming pool

Given route map is

A person can go from entrance to swimming pool in the following ways:

i.) Entrance to Gym to swimming pool i.e., E$\to$G$\to$S

ii.) Entrance to Indoor stadium to pool i.e., E$\to$I$\to$S

iii.)Entrance to Indoor stadium to Gym to swimming pool E$\to$I$\to$G$\to$S

iv.)Entrance to Gym to Indoor stadium to swimming pool E$\to$G$\to$I$\to$S

The number of ways of following each of the above paths is

i.) E$\to$G$\to$S can be done in $3\times 4$ = 12 ways

ii.) E$\to$I$\to$S can be done in $2\times 1$ = 2 ways

iii.) E$\to$I$\to$G$\to$S can be done in $2\times 1\times 4$ = 8 ways

iv.) E$\to$G$\to$I$\to$S can be done in $3\times 1\times 1$ = 3 ways

Now, according to the fundamental principle of counting,

Number of ways of going from entrance to swimming pool $=12+2+8+3=25$ ways