Question

Number of ways for a person $A$ to travel from $(0,0)$ to $(16,16)$ in the coordinate plane by moving one unit in the positive $x$ or $y$ direction at a time, such that $A$ changes direction an odd number of times, is equal to

• A. ${}^{30}{C}_{15}$
• B. $2\cdot {}^{30}{C}_{15}$
• C. ${}^{30}{C}_{14}\cdot {}^{30}{C}_{16}$
• D. ${}^{30}{C}_{15}\cdot {}^{30}{C}_{15}$

Answer 1

The correct option is B $2\cdot {}^{30}{C}_{15}$

Since $A$ changes direction an odd number of times, the first and the last step should be in different directions.
Case $\text{I}:$ If the first step is in $x$ direction and the last step is in $y$ direction, number of paths from $(1,0)$ to $(16,15)$ is ${}^{30}{C}_{15}$

Case $\text{II}:$ If the first step is in $y$ direction and the last step is in $x$ direction, number of paths from $(0,1)$ to $(15,16)$ is ${}^{30}{C}_{15}$
Total number of ways $={}^{30}{C}_{15}+{}^{30}{C}_{15}=2\cdot {}^{30}{C}_{15}$