1. **Problem statement:** How many 2-letter code words can be formed from the letters X, N, I, M, A, P under three conditions: - No letter is repeated. - Letters can be repeated. - Adjacent letters must be different. 2. **Total letters available:** There are 6 letters: X, N, I, M, A, P. --- ### Case 1: No letter is repeated 3. For the first letter, we can choose any of the 6 letters. 4. For the second letter, since no repetition is allowed, we have 5 letters left. 5. Using the multiplication principle: $$6 \times 5 = 30$$ So, there are 30 possible 2-letter code words with no repeated letters. --- ### Case 2: Letters can be repeated 6. For the first letter, 6 choices. 7. For the second letter, since repetition is allowed, again 6 choices. 8. Total number of code words: $$6 \times 6 = 36$$ --- ### Case 3: Adjacent letters must be different 9. This is similar to Case 1 because the only restriction is that the two letters are different. 10. For the first letter, 6 choices. 11. For the second letter, it cannot be the same as the first, so 5 choices. 12. Total number of code words: $$6 \times 5 = 30$$ --- **Final answers:** - No letter repeated: 30 - Letters can be repeated: 36 - Adjacent letters different: 30