Question

Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at (i, j) then it can move to either (i + 1, j) or (i, j + 1).
How many distinct paths are there for the robot to reach the point (10, 10) starting from the initial position (0, 0)?

• A. $\left(\begin{array}{c}20\\ 10\end{array}\right)$
• B. $2^{20}$
• C. $2^{20}$
• D. None of these

Answer 1

The correct option is A $\left(\begin{array}{c}20\\ 10\end{array}\right)$

Consider the following diagram:

The robot can move only right or up as defined in problem. Let us denote right move by 'R' and up move by `U'. Now to reach (3, 3) from (0, 0), the robot has to make exactly 3'R' moves and 3 'U' moves in any order.
Similarly to reach (10, 10) from (0, 0), the robot has to make 10 'R' moves and 10 'U' moves in any order. The number of ways this can be done is same as number of permutations of a word consisting of 10 'R's and 10 'U's.
Applying formula of permutation with limited repetitions we get the answer $as\frac{20!}{10!10!}$
${=}^{20}{C}_{10}$