how many permutations of "HOLIDAY" are there if "AD" adjacent and "LI" separated

To solve this permutation problem with restrictions, we need to consider both the adjacency of "AD" and the separation of "LI."

### Step-by-step solution:

1. **Total letters**:
   The word "HOLIDAY" has 7 distinct letters: H, O, L, I, D, A, Y.

2. **Treat "AD" as a single block**:
   Since "A" and "D" must be adjacent, we treat "AD" as one block. This reduces the total number of elements to permute from 7 to 6: H, O, L, I, Y, and the block "AD."

3. **Permutations of the blocks**:
   Now, there are \( 6! \) (6 factorial) ways to arrange these 6 elements.
   \[
   6! = 720
   \]

4. **Account for "LI" being separated**:
   To ensure that L and I are not adjacent, we use the total number of arrangements without the restriction and subtract the cases where L and I are adjacent.

   - If L and I are adjacent, treat "LI" as a block. Now, you have 5 elements to permute: H, O, the block "LI", Y, and the block "AD". This gives \( 5! \) arrangements:
     \[
     5! = 120
     \]
   
   - Subtract the cases where L and I are adjacent from the total number of unrestricted arrangements:
     \[
     \text{Valid arrangements} = 720 - 120 = 600
     \]

Thus, the number of permutations of "HOLIDAY" where "AD" is adjacent and "LI" is separated is **600**.

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What if "LI" were adjacent instead of separated?
2. How many permutations exist if "A" and "L" must also be adjacent?
3. What if both "AD" and "LI" must be adjacent?
4. Can you calculate the number of permutations without any restrictions?
5. How would the solution change if the letters in "HOLIDAY" weren't distinct?

**Tip**: For problems involving restrictions in permutations, breaking down the problem into smaller cases can simplify the process.

how many permutations of 'HOLIDAY' are there if 'AD' adjacent and 'LI' separated

Find out how many permutations of the word HOLIDAY exist when AD must be adjacent and LI must be separated. This combinatorics problem applies factorial calculations and the inclusion-exclusion principle to solve for 600 valid arrangements, making it a perfect problem for high school students.

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