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The problem shown asks for the number of direct routes from point \( A \) to point \( B \) on a grid. This is a classic combinatorial problem often solved using binomial coefficients.

### Explanation:

The grid shown is a 5x3 grid, meaning to get from point \( A \) to point \( B \), you need to make a total of 8 moves: 5 moves to the right (R) and 3 moves up (U).

The number of different paths is the number of distinct ways to arrange these 8 moves where 5 are right (R) and 3 are up (U). This can be calculated using the binomial coefficient:

\[
\text{Number of paths} = \binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
\]

So, the number of direct routes from point \( A \) to point \( B \) is **56**.

Would you like more details or have any questions about this solution?

### Related Questions:
1. How would the solution change if the grid were larger, say 6x4?
2. What is the formula for calculating the number of paths on a grid with arbitrary dimensions \( m \times n \)?
3. How can the problem be extended to grids where some paths are blocked?
4. What if diagonal moves (up-right) were allowed?
5. Can this problem be solved using dynamic programming? How?

### Tip:
Remember, combinatorial problems like this often use binomial coefficients, which count the number of ways to choose a subset of items from a larger set, essential in many grid path problems.

What is the number of direct routes from point A to point B on a 5x3 grid?

Find the number of direct routes from point A to point B on a 5x3 grid. This combinatorics problem involves calculating distinct paths using the binomial coefficient formula, suitable for Grades 9-12.

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