How many arrangements can be made using the following letter P Q C D P A Q P

To find how many distinct arrangements can be made using the letters **P, Q, C, D, P, A, Q, P**, we need to calculate the total permutations of these letters while accounting for repetitions of identical letters.

### Step 1: Total number of letters
The total number of letters is 8.

### Step 2: Identify repeated letters
- **P** appears 3 times
- **Q** appears 2 times
- **C**, **D**, and **A** each appear 1 time

### Step 3: Use the formula for permutations with repetition
The formula for permutations of multiset is:
\[
\text{Number of permutations} = \frac{n!}{k_1! \cdot k_2! \cdot \cdots \cdot k_r!}
\]
where:
- \(n\) is the total number of letters
- \(k_1, k_2, \dots, k_r\) are the frequencies of the repeated letters.

Substitute the values:
\[
\text{Number of permutations} = \frac{8!}{3! \cdot 2! \cdot 1! \cdot 1! \cdot 1!}
\]
### Step 4: Calculate factorials
- \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\)
- \(3! = 3 \times 2 \times 1 = 6\)
- \(2! = 2 \times 1 = 2\)

Now, compute:
\[
\text{Number of permutations} = \frac{40,320}{6 \times 2 \times 1 \times 1 \times 1} = \frac{40,320}{12} = 3,360
\]

### Final Answer:
There are **3,360 distinct arrangements** of the letters **P, Q, C, D, P, A, Q, P**.

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Do you want more details or have any questions?

Here are some related questions to expand your understanding:
1. How would the formula change if there were more repeated letters?
2. What if you were asked to arrange only a subset of the letters?
3. How many arrangements would there be if all letters were distinct?
4. What if we were asked to arrange the letters in a circular arrangement?
5. How does the number of distinct letters affect the number of arrangements?

**Tip:** When dealing with permutations, always account for any repeated items to avoid overcounting!

Explore how to calculate the number of distinct arrangements of the letters P Q C D P A Q P, using permutations with repetition. This solution includes a detailed breakdown of factorial calculations and the permutations formula, suitable for students in Grades 10-12.

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