Final answer:
To find the number of distinct paths a king can take to travel from the left bottom square (1,1) to the top right square (n,m), we can use the concept of combinations.
Step-by-step explanation:
To find the number of distinct paths a king can take to travel from the left bottom square (1,1) to the top right square (n,m), we can use the concept of combinations. Since the king can only move either vertically or horizontally on each move, we can calculate the number of distinct paths using the formula for combinations.
Let's assume the king needs to move n steps horizontally and m steps vertically. The total number of moves required to reach the destination is n + m . To find the number of distinct paths, we need to choose either the n steps horizontally or the m steps vertically. We can do this by calculating the combinations of choosing either n or m steps from a total of n + m steps.
The formula for combinations is given by:
C(n + m, n) = (n + m)! / (n!(m!))
Therefore, the number of distinct paths a king can take is C(n + m, n).