help

We are asked to determine how many distinct paths can be followed to spell the word "MATH" starting from the origin \( M \), given that movements are only allowed up, down, left, and right.

The points corresponding to \( A \), \( T \), and \( H \) are labeled on the xy-plane, and each of these labels corresponds to a specific set of coordinates.

### Step 1: Understanding the Problem

The problem provides:

- \( M \) is at the origin, \( (0, 0) \).

- \( A \)'s are at \( (1,0) \), \( (-1,0) \), \( (0,1) \), and \( (0,-1) \).

- \( T \)'s are at \( (2,0) \), \( (1,1) \), \( (0,2) \), \( (-1,1) \), \( (-2,0) \), \( (-1,-1) \), \( (0,-2) \), and \( (1,-1) \).

- \( H \)'s are at \( (3,0) \), \( (2,1) \), \( (1,2) \), \( (0,3) \), \( (-1,2) \), \( (-2,1) \), \( (-3,0) \), \( (-2,-1) \), \( (-1,-2) \), \( (0,-3) \), \( (1,-2) \), and \( (2,-1) \).

We need to determine how many distinct paths can be followed to spell "MATH", moving from \( M \) to an \( A \), then from \( A \) to a \( T \), and finally from \( T \) to an \( H \).

### Step 2: Movement Considerations

We are allowed to move only up, down, left, and right. This restricts the possible movements between the points labeled \( M \), \( A \), \( T \), and \( H \).

- From \( M \) at \( (0, 0) \), we can move to any of the \( A \)'s at \( (1,0) \), \( (-1,0) \), \( (0,1) \), or \( (0,-1) \).

- From each \( A \), we can move to one of the \( T \)'s that are one unit away from the \( A \)'s.

- From each \( T \), we can move to one of the \( H \)'s that are one unit away from the \( T \)'s.

### Step 3: Counting the Distinct Paths

Let’s break down the path counting process step by step.

#### Paths from \( M \) to \( A \):

From \( M = (0, 0) \), there are 4 possible \( A \)'s:

- \( A_1 = (1, 0) \)

- \( A_2 = (-1, 0) \)

- \( A_3 = (0, 1) \)

- \( A_4 = (0, -1) \)

So, there are 4 choices for the first step.

#### Paths from \( A \) to \( T \):

From each \( A \), we can move to a neighboring \( T \) that is one unit away. Let's examine the options for each \( A \):

- From \( A_1 = (1, 0) \), the possible \( T \)'s are \( (2, 0) \), \( (1, 1) \), and \( (1, -1) \). This gives 3 choices.

- From \( A_2 = (-1, 0) \), the possible \( T \)'s are \( (-2, 0) \), \( (-1, 1) \), and \( (-1, -1) \). This gives 3 choices.

- From \( A_3 = (0, 1) \), the possible \( T \)'s are \( (0, 2) \), \( (1, 1) \), and \( (-1, 1) \). This gives 3 choices.

- From \( A_4 = (0, -1) \), the possible \( T \)'s are \( (0, -2) \), \( (1, -1) \), and \( (-1, -1) \). This gives 3 choices.

Thus, for each \( A \), there are 3 possible \( T \)'s, so the total number of ways to move from \( A \) to \( T \) is \( 3 \times 4 = 12 \).

#### Paths from \( T \) to \( H \):

From each \( T \), we can move to a neighboring \( H \) that is one unit away. There are 3 neighboring \( H \)'s for each \( T \) (similarly to the calculation above). Therefore, for each \( T \), there are 3 possible \( H \)'s.

Thus, for each \( T \), there are 3 possible \( H \)'s, so the total number of ways to move from \( T \) to \( H \) is \( 3 \times 12 = 36 \).

### Step 4: Total Number of Paths

Multiplying the number of choices at each step, we get:

\[
4 \times 3 \times 3 = 36.
\]

Thus, the total number of distinct paths that can be followed to spell the word "MATH" is \( \boxed{36} \).