The problem is to determine the number of distinct arrangements of the letters in the word REINTERPRET, accounting for identical letters.

Step 1: Count the occurrences of each letter

The word REINTERPRET has 11 letters in total. The frequencies of the letters are:

• R: 3 occurrences
• E: 3 occurrences
• I: 1 occurrence
• N: 1 occurrence
• T: 2 occurrences
• P: 1 occurrence

Step 2: Apply the formula for permutations with repeated letters

The formula for the number of distinct permutations of a multiset is:

$\text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}$

where:

• $n$ is the total number of items (11 in this case),
• $n_1, n_2, \ldots, n_k$ are the frequencies of each repeated item.

Here: $n = 11, \quad n_R = 3, \quad n_E = 3, \quad n_I = 1, \quad n_N = 1, \quad n_T = 2, \quad n_P = 1$

Substitute into the formula: $\text{Number of arrangements} = \frac{11!}{3! \cdot 3! \cdot 1! \cdot 1! \cdot 2! \cdot 1!}$

Step 3: Calculate the factorials

• $11! = 39916800$
• $3! = 6$
• $2! = 2$
• $1! = 1$

Substitute these values: $\text{Number of arrangements} = \frac{39916800}{6 \cdot 6 \cdot 1 \cdot 1 \cdot 2 \cdot 1}$

Simplify: $\text{Number of arrangements} = \frac{39916800}{72} = 554400$

Final Answer:

The number of distinct arrangements of the letters in REINTERPRET is 554,400.

---

Would you like further explanation or assistance with related problems? Here are some related questions for practice:

1. How many distinct arrangements can be made from the word "INTERPRET"?
2. If one additional "E" is added to "REINTERPRET", how many arrangements are possible?
3. What is the probability of randomly arranging "REINTERPRET" and having "R" as the first letter?
4. How does the result change if all letters are distinct in a word with 11 characters?
5. In how many ways can the letters in "REINTERPRET" be arranged if all "R"s must appear consecutively?

Tip: Always check the counts of each repeated letter carefully before applying the formula.