https://file.aiquickdraw.com/mathbot/file/172541274891463xm4fhx.jpg

The problem presented asks how many direct routes Tyler can take from point A to point B on the grid shown, moving along the streets without backtracking. This type of problem can be solved using the concept of combinatorics, specifically calculating the number of possible paths on a grid.

### Step-by-Step Solution:

1. **Understanding the Grid Layout:**
   - Tyler is moving from Point A to Point B on a grid.
   - To reach from Point A to Point B, Tyler must move a certain number of blocks horizontally (let's say \( x \)) and a certain number of blocks vertically (let's say \( y \)).

2. **Counting the Number of Steps:**
   - To determine the number of routes, first, count the number of horizontal and vertical steps needed.
   - From the image:
     - Horizontal steps (Eastward): 4 blocks.
     - Vertical steps (Southward): 3 blocks.
   
3. **Calculating the Number of Routes:**
   - The number of routes Tyler can take is given by the combination formula:
   \[
   \text{Number of routes} = \binom{x+y}{x} = \binom{7}{4}
   \]
   - Here, \( \binom{7}{4} \) represents the number of ways to choose 4 eastward moves out of the total 7 moves.

4. **Computing the Combination:**
   - Calculate \( \binom{7}{4} \):
   \[
   \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35
   \]
   - So, there are 35 possible routes that Tyler can take without backtracking.

### Conclusion:
Tyler can take **35 direct routes** from Point A to Point B without backtracking.

Would you like any further details on this solution?

Here are five related questions to expand on this concept:

1. How would the number of routes change if Tyler had to move 5 blocks eastward and 4 blocks southward?
2. What if there were obstacles on the grid that Tyler had to avoid? How would you calculate the number of routes?
3. How many routes would there be if Tyler was allowed to backtrack?
4. How can this problem be extended to a 3D grid with vertical movement allowed?
5. What are some real-life applications of calculating routes in a grid?

**Tip:** The concept of combinatorial paths on a grid is closely related to Pascal's Triangle, where each number represents the number of ways to get to a point on the grid.

Tyler wishes to walk along the streets from Point A to Point B. How many direct routes (no backtracking) can Tyler take?

Learn how to calculate the number of direct routes Tyler can take from Point A to Point B on a grid. This problem involves combinatorial concepts and uses the combination formula to find the solution. Suitable for students in Grades 8-10.

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