I am looking for help to this question
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?
pls give answer
The answer is 28 :)
To solve this problem, we need to find all the distinct paths that can be followed to spell out the word MATH by moving up, down, left, and right starting from the origin.
Let's start by looking at the word MATH and the respective points associated with each letter.
M: Origin (0, 0)
A: Points (1, 0), (-1, 0), (0, 1), (0, -1)
T: Points (2, 0), (1, 1), (0, 2), (-1, 1), (-2, 0), (-1, -1), (0, -2), (1, -1)
H: Points (3, 0), (2, 1), (1, 2), (0, 3), (-1, 2), (-2, 1), (-3, 0), (-2, -1), (-1, -2), (0, -3), (1, -2), (2, -1)
To find the number of distinct paths, we can use a recursive approach:
1. Start at the origin point (0, 0).
2. Move to one of the neighboring points labeled A (up, down, left, or right).
3. Repeat step 2 for the next letter in the word MATH.
4. When we reach the last letter, we have found a distinct path.
5. Backtrack to the previous letter and explore other paths until all possible paths are exhausted.
Let's go through the process step by step:
1. Starting at the origin (0, 0) labeled with M, we can move to any neighboring point labeled A (1, 0), (-1, 0), (0, 1), or (0, -1).
2. For each of these points, we can move to any neighboring point labeled T.
3. For each point labeled T, we can move to any neighboring point labeled H.
4. Finally, for each point labeled H, we have reached the end of the word MATH.
As we explore all possible paths, we need to make sure we avoid revisiting a point we have already visited to ensure distinct paths.
By following this recursive approach, we can count the total number of distinct paths until we reach the end of the word MATH.
Note that there might be different ways to implement this solution, such as using dynamic programming or backtracking techniques, but the overall approach remains the same.
I hope this explanation helps you understand how to approach the problem.
Ok, ill get rid of them, I graphed it and saw nothing
well, start small.
Starting at each of the M's, how many ways get you to an A?
get out some graph paper.
Plot and label all the points.
Draw a line from each M to any A one unit away.
Continue on to the T's, and H's.
Do something!
Start with short paths, extend, and watch for patterns that develop. What do you get?
And lose all the $$ signs, OK? They just make things hard to read.