Question

A postman can travel from village A to village B by 4 different roads , from village B to village C by 2 different roads and from village C to village D by 3 different roads .

If a postman goes from village A to village D and comes back by a different route ,in how many ways can he do this ?

Answer 1

Total number of ways from A to D is 4*2*3=24. No of ways of coming back is 3*1*2(excluding the routes taken from A to D). Hence total number of ways is 24*6=144.

Answer 2

NUmber of ways of going from A to D are : multiplication of number of ways of going from A to B, and number of ways of going from B to C,  and the number of ways of going from C to D....

This is because each of the routes or ways of travelling is independent of others.

So ways of going from A to D are  4 * 2 * 3 = 24

Now, on the way return from D to A, If  no part of the route is to be traced back, then the possible ways left are  2 from D to C,  1 from C to B,  and 3 from B to A.

Hence  number of ways of choosing new routes on the return are :
               2 * 1 * 3 = 6.

The total number of ways of going from A to D and  returning back to A, are :
             24 * 6  = 144.
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This answer will change if a part of the route can be traced back.. like A to B or B to C etc..  That number will be much more.