create a tree diagram to show all possible arrangements of the word "MATH" how many outcomes exist?
To create a tree diagram for all possible arrangements of the word "MATH," we can start by listing all the letters of the word.
1st level: The first letter can be either "M," "A," "T," or "H."
2nd level: Under each letter in the 1st level, list the remaining letters.
- If the 1st letter is "M," the 2nd letter can be either "A," "T," or "H."
- If the 1st letter is "A," the 2nd letter can be either "M," "T," or "H."
- If the 1st letter is "T," the 2nd letter can be either "M," "A," or "H."
- If the 1st letter is "H," the 2nd letter can be either "M," "A," or "T."
3rd level: Under each letter in the 2nd level, list the remaining letters.
- Repeat this process until all letters are placed in each branch.
The completed tree diagram would look like this:
```
M
________|_________
| | |
A T H
_____|____ _|___ ___|___
| | | | | | | |
T H T A H A T M
| | | | | | | |
H T H T T M A H
| | | | | | |
T H A M M T A
| | | |
A M H H
|
H
```
To determine the number of outcomes, we count the number of leaves in the tree diagram. In this case, there are a total of 24 leaves, which represents the number of possible arrangements of the word "MATH."