Consider the Cartesian (x-y) grid From A(0, 0) to C(9, 10) if you must pass through D(5, 4) and E(7, 8)?

To solve this problem, we need to determine the number of paths from point \( A(0, 0) \) to \( C(9, 10) \) while passing through points \( D(5, 4) \) and \( E(7, 8) \).

### Step-by-Step Solution:
We will break the problem into three segments:
1. From \( A(0, 0) \) to \( D(5, 4) \).
2. From \( D(5, 4) \) to \( E(7, 8) \).
3. From \( E(7, 8) \) to \( C(9, 10) \).

We calculate the number of paths for each segment and then multiply these numbers together, as each segment is independent.

#### 1. Paths from \( A(0, 0) \) to \( D(5, 4) \):
- You need to move 5 steps right (R) and 4 steps up (U).
- The total number of paths is given by the binomial coefficient:

\[
\text{Paths from } A \text{ to } D = \binom{5+4}{4} = \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
\]

#### 2. Paths from \( D(5, 4) \) to \( E(7, 8) \):
- You need to move 2 steps right (R) and 4 steps up (U).
- The total number of paths is:

\[
\text{Paths from } D \text{ to } E = \binom{2+4}{4} = \binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15
\]

#### 3. Paths from \( E(7, 8) \) to \( C(9, 10) \):
- You need to move 2 steps right (R) and 2 steps up (U).
- The total number of paths is:

\[
\text{Paths from } E \text{ to } C = \binom{2+2}{2} = \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
\]

### Total Number of Paths:
To find the total number of paths from \( A \) to \( C \) through \( D \) and \( E \), multiply the number of paths for each segment:

\[
\text{Total Paths} = 126 \times 15 \times 6 = 11340
\]

### Conclusion:
The total number of paths from \( A(0, 0) \) to \( C(9, 10) \) passing through \( D(5, 4) \) and \( E(7, 8) \) is **11,340**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. What if the points \( D(5, 4) \) and \( E(7, 8) \) were swapped in the sequence of visits?
2. How many paths exist from \( A(0, 0) \) to \( C(9, 10) \) without any intermediate points?
3. What if you added another intermediate point \( F(8, 9) \) that must also be passed?
4. How does the problem change if you are allowed to move left and down as well?
5. Can you find a general formula for the number of paths through multiple intermediate points?

**Tip:** When breaking down complex path problems, always consider the independence of segments, which allows you to calculate each segment separately and then combine them.

Learn how to calculate the number of paths from point A(0, 0) to point C(9, 10) passing through intermediate points D(5, 4) and E(7, 8). Detailed step-by-step solution using combinatorics and binomial coefficients. Suitable for Grades 10-12.

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