A toad is trying to get from point A (0, 0) to point B (13, 4) without overshooting, but he can only move up or right. He MUST alternate his step sizes between moving 1 unit in a single direction and moving 3 units in a single direction. Compute the number of ways the toad can start at point A and end at point B.

To solve this problem, we need to determine the number of distinct paths a toad can take from point \( A(0,0) \) to point \( B(13,4) \), moving only right or up. The key constraint is that the toad must alternate between moving 1 unit in a direction and 3 units in a direction.

### Step 1: Understand the problem constraints
- **Rightward moves**: The toad must move from \( x = 0 \) to \( x = 13 \). The total horizontal distance to be covered is 13 units.
- **Upward moves**: The toad must move from \( y = 0 \) to \( y = 4 \). The total vertical distance to be covered is 4 units.
- The toad alternates between 1-unit moves and 3-unit moves.

### Step 2: Define the possible moves
- **Rightward move of 1 unit**: We denote this as \( R_1 \).
- **Rightward move of 3 units**: We denote this as \( R_3 \).
- **Upward move of 1 unit**: We denote this as \( U_1 \).
- **Upward move of 3 units**: We denote this as \( U_3 \).

### Step 3: Set up the equation for the number of moves
- Let \( x_1 \) be the number of \( R_1 \) moves and \( x_3 \) be the number of \( R_3 \) moves.
- Let \( y_1 \) be the number of \( U_1 \) moves and \( y_3 \) be the number of \( U_3 \) moves.

We have the following system of equations:

\[
x_1 + 3x_3 = 13 \quad \text{(rightward distance)}
\]
\[
y_1 + 3y_3 = 4 \quad \text{(upward distance)}
\]

### Step 4: Find the solutions
1. **Rightward moves**:
   - \( x_1 + 3x_3 = 13 \)
   - Possible pairs \( (x_1, x_3) \) are:
     - \( x_3 = 0 \), \( x_1 = 13 \)
     - \( x_3 = 1 \), \( x_1 = 10 \)
     - \( x_3 = 2 \), \( x_1 = 7 \)
     - \( x_3 = 3 \), \( x_1 = 4 \)
     - \( x_3 = 4 \), \( x_1 = 1 \)

2. **Upward moves**:
   - \( y_1 + 3y_3 = 4 \)
   - Possible pairs \( (y_1, y_3) \) are:
     - \( y_3 = 0 \), \( y_1 = 4 \)
     - \( y_3 = 1 \), \( y_1 = 1 \)

### Step 5: Calculate the number of ways
For each valid combination of \( (x_1, x_3) \) and \( (y_1, y_3) \), the total number of moves is \( x_1 + x_3 + y_1 + y_3 \). The number of distinct paths is determined by the multinomial coefficient:

\[
\text{Number of ways} = \frac{(x_1 + x_3 + y_1 + y_3)!}{x_1! \cdot x_3! \cdot y_1! \cdot y_3!}
\]

Let's compute the number of paths for each combination:

1. \( (x_1, x_3) = (13, 0) \) and \( (y_1, y_3) = (4, 0) \)
   - Total steps: \( 13 + 0 + 4 + 0 = 17 \)
   - Paths: \( \frac{17!}{13! \cdot 4!} \)

2. \( (x_1, x_3) = (10, 1) \) and \( (y_1, y_3) = (4, 0) \)
   - Total steps: \( 10 + 1 + 4 + 0 = 15 \)
   - Paths: \( \frac{15!}{10! \cdot 1! \cdot 4!} \)

3. \( (x_1, x_3) = (7, 2) \) and \( (y_1, y_3) = (4, 0) \)
   - Total steps: \( 7 + 2 + 4 + 0 = 13 \)
   - Paths: \( \frac{13!}{7! \cdot 2! \cdot 4!} \)

4. \( (x_1, x_3) = (4, 3) \) and \( (y_1, y_3) = (4, 0) \)
   - Total steps: \( 4 + 3 + 4 + 0 = 11 \)
   - Paths: \( \frac{11!}{4! \cdot 3! \cdot 4!} \)

5. \( (x_1, x_3) = (1, 4) \) and \( (y_1, y_3) = (4, 0) \)
   - Total steps: \( 1 + 4 + 4 + 0 = 9 \)
   - Paths: \( \frac{9!}{1! \cdot 4! \cdot 4!} \)

6. \( (x_1, x_3) = (13, 0) \) and \( (y_1, y_3) = (1, 1) \)
   - Total steps: \( 13 + 0 + 1 + 1 = 15 \)
   - Paths: \( \frac{15!}{13! \cdot 1! \cdot 1!} \)

7. \( (x_1, x_3) = (10, 1) \) and \( (y_1, y_3) = (1, 1) \)
   - Total steps: \( 10 + 1 + 1 + 1 = 13 \)
   - Paths: \( \frac{13!}{10! \cdot 1! \cdot 1! \cdot 1!} \)

8. \( (x_1, x_3) = (7, 2) \) and \( (y_1, y_3) = (1, 1) \)
   - Total steps: \( 7 + 2 + 1 + 1 = 11 \)
   - Paths: \( \frac{11!}{7! \cdot 2! \cdot 1! \cdot 1!} \)

9. \( (x_1, x_

Learn how to count the number of distinct paths a toad can take from point A(0,0) to point B(13,4), alternating step sizes (1 unit or 3 units) while moving only right or up. This problem involves combinatorial counting using multinomial coefficients.

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