### Instruction: On the $xy$-plane, the origin is labeled with an $M$. The points $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$ are labeled with $A$'s. The points $(2,0)$, $(1,1)$, $(0,2)$, $(-1, 1)$, $(-2, 0)$, $(-1, -1)$, $(0, -2)$, and $(1, -1)$ are labeled with $T$'s. The points $(3,0)$, $(2,1)$, $(1,2)$, $(0, 3)$, $(-1, 2)$, $(-2, 1)$, $(-3, 0)$, $(-2,-1)$, $(-1,-2)$, $(0, -3)$, $(1, -2)$, and $(2, -1)$ are labeled with $H$'s. If you are only allowed to move up, down, left, and right, starting from the origin, how many distinct paths can be followed to spell the word MATH?### Response: I want to count the number of ways to go from M to A, then from A to T, then from T to H, and then multiply them all together.
To go from M to A, I have four choices: up, down, left, or right.
To go from A to T, I have to be careful not to repeat the A I just came from.
If I went up from M to A, then I can go up, left, or right from A to T, but not down.
That's three choices.
Similarly, if I went down from M to A, then I can go down, left, or right from A to T, but not up.
That's also three choices.
If I went left from M to A, then I can go up, down, or left from A to T, but not right.
That's also three choices.
If I went right from M to A, then I can go up, down, or right from A to T, but not left.
That's also three choices.
So no matter which way I go from M to A, I have three choices to go from A to T. That means the total number of ways to go from M to A and then to T is 4 times 3, or 12.
To go from T to H, I have to be careful not to repeat the T I just came from.
If I went up from A to T, then I can go up, left, or right from T to H, but not down.
That's three choices.
Similarly, if I went down from A to T, then I can go down, left, or right from T to H, but not up.
That's also three choices.
If I went left from A to T, then I can go up, down, or left from T to H, but not right.
That's also three choices.
If I went right from A to T, then I can go up, down, or right from T to H, but not left.
That's also three choices.
So no matter which way I go from A to T, I have three choices to go from T to H. That means the total number of ways to go from A to T and then to H is 12 times 3, or 36.
Therefore, the total number of ways to go from M to A to T to H is 4 times 36, or 144.