To solve this problem, we need to determine the number of ways a person can walk from point A to point C, passing through point B, while only moving rightwards or upwards. The grid shown is a 5x5 grid, with points A, B, and C located as follows:

• $A$ is at the bottom-left corner (0, 0).
• $B$ is at the center (3, 2).
• $C$ is at the top-right corner (5, 5).

Steps:

1. From A to B:

   • To move from $A(0, 0)$ to $B(3, 2)$, the person needs to make exactly 3 rightward moves and 2 upward moves.
   • The number of ways to arrange these 3 rightward and 2 upward moves is given by the number of permutations of these 5 moves (3 rightward, 2 upward): $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10$ So, there are 10 ways to go from A to B.

2. From B to C:

   • To move from $B(3, 2)$ to $C(5, 5)$, the person needs to make exactly 2 rightward moves and 3 upward moves.
   • The number of ways to arrange these 2 rightward and 3 upward moves is given by the number of permutations of these 5 moves (2 rightward, 3 upward): $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$ So, there are 10 ways to go from B to C.

Total Number of Ways:

Since the two parts (A to B and B to C) are independent, the total number of ways is the product of the two: $10 \times 10 = 100$

Final Answer:

There are 100 ways to walk from A to C via B.

Do you want more details on any of the steps? Feel free to ask!

Here are five related questions to explore further:

1. How do you calculate the number of combinations for any two points on a grid?
2. What if diagonal movements were allowed? How would that change the calculation?
3. What if the grid was larger (e.g., 7x7)? How would you calculate the number of ways from A to C via B?
4. How does the problem change if obstacles are placed on the grid?
5. How would you calculate the shortest path if the grid was not square?

Tip: The number of ways to move in a grid is typically based on choosing how many moves to make in each direction, often calculated with combinations (binomial coefficients).