IIT JEE , AIEEE Forum - Mathematics , Algebra

Mr.IITIAN007

There are n₁streets in a city running N-S and n₂ streets running E-W.Then

what is the number of ways in which a person can travel from one corner to the diagonally opposite corner through the

i)shortest possible distance ?

ii)longest possible distance

avinash.sharma

i) As we conclude from the figure (given below) that to cross diagonally one has to cross n₁-1 roads N-S and n₂-1 roads from E-W. So he has to select n₁-1 roads N-S from n₁+n₂-2 and n₂-1 roads E-W from n₁+n₂-2 so the required ways are

(^n1+n2-2 C_n1-1)* (^n1+n2-2 C_n2-1) ways

ii) as indicated in the figure, it is clear that the longest distance can be covered only by two ways either starts with N-S direction or with E-W direction and depend on the length of n1 and n2 roads.

Suppose x is the side of a small rectangle in N-S direction and y is the length of the side of a rectangle in E-W direction then from the figure you can observe that one has to move 20 steps in N-S direction and 5 steps in E-W direction so total distance covered by him is 20x+5y (2 ways are possible and marked in the figure in different colors) but if starts his journey from E-W direction than he has to travel 25 steps in E-W direction and only 4 steps in N-S direction so distance travel by him 4x + 25 y so the distance traveled is depends on length of roads of N1 and N2 (there are again 2 ways possible to travel in this method, left for you imagination).

amar.gupta

I also agree with Mr. Avinash Sharma

Mr.IITIAN007

I too agree with you Mr.Avinash Sharma . Thank you.

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