mynhauzen wrote:

Two highways start from a point P and meet again at the point Q. Between P and Q there are 5 subways, joining the highways at 5 different places. How many different routes are possible for a journey from P to Q?

(A) 12

(B) 16

(C) 24

(D) 32

(E) 64

not sure, how to work with this kind of problem

No. of ways of using no subways = \(2*5C_0 = 2*1 = 2\)

No. of ways of using one subway = \(2*5C_1 = 2*5 = 10\)

No. of ways of using two subways = \(2*5C_2 = 2*10 = 20\)

No. of ways of using three subways = \(2*5C_3 = 2*10 = 20\)

No. of ways of using four subways = \(2*5C_4 = 2*5 = 10\)

No. of ways of using five subways = \(2*5C_5 = 2*1 = 2\)

We are multiplying by 2 on each occasion because for each combination, we can start the trip either by route 1 or by route 2.

Total = 2 + 10 + 20 + 20 + 10 + 2 = 64

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