How many 3-letter codes can be formed with the letters A, B, and C if each letter can
only be used once in the code? Draw a tree diagram.

To solve this problem, we want to determine how many distinct 3-letter codes can be formed from the letters A, B, and C, where no letter is repeated.

### Steps:

1. **First letter**: Since each letter can only be used once, you have 3 choices (A, B, or C) for the first letter.
2. **Second letter**: After choosing the first letter, there are 2 remaining choices for the second letter.
3. **Third letter**: After the first two letters are chosen, only 1 letter remains for the third position.

### Calculation:
- For the first letter, there are 3 possibilities.
- For the second letter, there are 2 possibilities (since one letter has already been used).
- For the third letter, there is only 1 possibility (since two letters have been used).

So, the total number of 3-letter codes is:
\[
3 \times 2 \times 1 = 6
\]

### Tree Diagram Explanation:

To visualize this, you can draw a **tree diagram** that branches out based on your choices for each position in the 3-letter code.

1. **First branch**: Start with the first letter (A, B, or C). This gives you 3 branches.
2. **Second branch**: From each of these branches, choose the second letter (only 2 remaining letters). Each branch splits into 2 sub-branches.
3. **Third branch**: Finally, choose the third letter (only 1 remaining). Each sub-branch leads to a single final letter.

I'll describe a rough breakdown of how the tree looks:

- **Root**: The beginning of the tree, representing the choice of the first letter.
    - **A branch**: From A, you have two options: B or C.
        - AB: From here, only C remains → ABC.
        - AC: From here, only B remains → ACB.
    - **B branch**: From B, two options remain: A or C.
        - BA: From here, only C remains → BAC.
        - BC: From here, only A remains → BCA.
    - **C branch**: From C, two options remain: A or B.
        - CA: From here, only B remains → CAB.
        - CB: From here, only A remains → CBA.

Here’s the complete set of possible codes:
- ABC
- ACB
- BAC
- BCA
- CAB
- CBA

Would you like me to generate the tree diagram for you?

How many 3-letter codes can be formed with the letters A, B, and C if each letter can only be used once in the code? Draw a tree diagram.

Learn how to form 3-letter codes using the letters A, B, and C without repetition. This problem covers combinatorics and permutations and includes a step-by-step solution along with a tree diagram. Suitable for students in Grades 5-7.

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