A(1, 2) and B(5, 5) are two points. Starting from A, line segments of unit length are drawn either rightwards or upwards only, in each step, until B is reached. Then, the number of ways of connecting A and B in this manner is

Given A (1, 2) and B (5, 5). Difference of x coordinates = 5 – 1 =4
$\therefore$  Exactly 4 rightward steps are needed. 
Difference of y coordinates = 5 – 2 = 3
$\therefore$  Exactly 3 upward steps are needed.
Note : Order of the steps is immaterial.
Denote each rightward step by R and each upward step by U.
$\therefore$  Product is arranging the letters RRRRUUU
No. of arrangements  $=\frac{7!}{4!3!}=35$