how many different paths in the xy-plane are there from (1,3) to (5,6) if a path proceeds one step at a time by going either one step to the right R or one step upward U?

ans is 35, but explain please.

In order to reach from (1,3) to (5,6) , We have to definitely 4 Steps UP (U), ans 3 Steps toward right (R).

So The order can be like

1. UUUURRR

2. UUURRRU

... 

This problem can be viewed as in how many ways one can arrange 4U and 3R together. the answer is 7!/(3!)(4!) as U and R are similar.

So the answer is 35

Approved

how u came to know that we have to move 3 steps right??

first of all...

i think we have to move 4steps to right and 3steps to upward if we imagine the diagram of xy-plane and poin (1,3) and (5.6)

there on axis.. we divide 7! by 4! and 3! this is cool... thankuu

Concept of vectors. You divide the position vector into horizontal and vertical components.