In how many different ways can the letters of the word HAPPYHOLI be arranged, taking all letters each time and considering that some letters are repeated?

Introduction / Context:
This question involves counting permutations of letters in a word that contains repeated letters. It checks whether the student can correctly apply the permutation formula for multisets. Accurate counting of repeated letters is crucial to avoid overcounting arrangements that look the same because some letters are identical.

Given Data / Assumptions:

• The word is HAPPYHOLI.
• The letters are H, A, P, P, Y, H, O, L, I.
• Total number of letters: 9.
• Repeated letters: H appears twice and P appears twice.
• We consider all arrangements of these 9 letters, counting only distinct permutations.

Concept / Approach:
If all letters were distinct, the total number of permutations would be 9!. However, because some letters are repeated, permuting identical letters among themselves does not create new distinct arrangements. To correct for this overcounting, we divide by the factorial of the number of times each letter is repeated. Therefore, the number of distinct permutations is 9! / (2! × 2!).

Step-by-Step Solution:
Step 1: Count total letters: there are 9 positions to fill, one for each letter of HAPPYHOLI.Step 2: Identify repetitions: the letter H appears 2 times, and the letter P appears 2 times. All other letters A, Y, O, L and I appear once.Step 3: Apply the permutation formula for repeated letters: total distinct arrangements = 9! / (2! × 2!).Step 4: Compute 9! = 362880.Step 5: Compute 2! × 2! = 2 × 2 = 4.Step 6: Divide 362880 by 4 to get 90720.Step 7: Therefore, the number of distinct arrangements of the letters of HAPPYHOLI is 90720.
Verification / Alternative check:
A quick reasonableness check is to compare with the case where all letters are distinct. That would give 362880 arrangements. Since two letters are repeated twice, we expect a reduction by a factor of 4 (2! for H and 2! for P), giving around 90000 arrangements. Our exact value 90720 fits this expectation perfectly, confirming the calculation.

Why Other Options Are Wrong:
Values like 72000 or 81000 correspond to different divisors or mistaken counts of repeated letters. 89772 or other similar numbers do not match 9! divided by any simple product of factorials of frequencies. Only 90720 equals 9! / (2! × 2!) and correctly reflects the presence of two repeated letters in the word.

Common Pitfalls:
Students sometimes miscount the number of times each letter appears, for example, noticing the two P's but missing that H also appears twice. Another common error is to use 9! directly without dividing by the factorials of repeated letters, thereby overcounting. Carefully writing out the letter frequencies before applying the formula is a good habit for such problems.

Final Answer:
The number of distinct arrangements of the letters of HAPPYHOLI is 90,720.