Question

# There are 4 routes between Delhi and Patna. In how many different ways can a man go from Delhi to Patna and return, if for returning (i) any of the routes is taken; (ii) the same route is taken ; (iii) the same route is not taken ?

Solution

## We have the following three cases. Case (i) When any of the routes is taken for returning : The man may take any route for going from Delhi to Patna. So, there are 4 ways of going from Delhi to Patna. When done so, he may return by any of the 4 routes. So, there are 4 ways of returning from Patna to Delhi. Hence, by the fundamental principle of multiplication, the total number of ways for going to Patna and returning back to Delhi = (4×4)=16. Case (ii) When the same route is taken for returning : In this case, there are 4 ways of going to Patna and only 1 way of returning, namely by the same route. Hence, the required number of ways = (4×1)=4. Case (iii) When the same route is not taken for returning : In this case, there are 4 ways of going to Patna. But, the man does not return by the same route. So, there are 3 ways of returning back to Delhi. Hence, the required number of ways = (4×3)=12.

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