# Answer to Question #30241 in Discrete Mathematics for anil jha

Question #30241

A man standing at a point P on a flat plane starts walking. At eacj step, he walks exactly 1 foot in one of the directions N,S,E or W. Suppose that after 6 steps he comes back to P, then no. of disctinct paths he can take is ?

Expert's answer

Let'sdivide 6 steps into 2 cases - first 3 and last 3 for him to be back at original

position, there should be 3 counter steps

E and W are counter

N and S are counterfor 1st 3 steps he has 4 options of each step so 4*4*4 =64now for last 3 steps, we need 3 counter steps the 3 counter steps can be at any position

so 3! i.e. 6

Then 64*6 = 384there are 4 ways in which he reaches origin from 1,0 which is not purely

through the axes hence 4*4 = 16 common cases

after 384, should we add 16

384 + 16 = 400Answer: 400

position, there should be 3 counter steps

E and W are counter

N and S are counterfor 1st 3 steps he has 4 options of each step so 4*4*4 =64now for last 3 steps, we need 3 counter steps the 3 counter steps can be at any position

so 3! i.e. 6

Then 64*6 = 384there are 4 ways in which he reaches origin from 1,0 which is not purely

through the axes hence 4*4 = 16 common cases

after 384, should we add 16

384 + 16 = 400Answer: 400

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