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?