In how many ways can the letters of the word PERMUTATIONS be arranged if  there are always 4 letters between P and S?

We are asked to find how many ways the letters of the word **PERMUTATIONS** can be arranged such that there are always 4 letters between the letters "P" and "S".

### Step 1: Analyze the word
The word **PERMUTATIONS** consists of 12 letters in total. However, some letters are repeated:
- P occurs once
- E occurs once
- R occurs once
- M occurs once
- U occurs once
- T occurs twice
- A occurs once
- I occurs once
- O occurs once
- N occurs once
- S occurs once

### Step 2: Define the positions of P and S
We are tasked with ensuring that there are always exactly 4 letters between **P** and **S**. To achieve this, let's calculate the number of possible ways to place **P** and **S** in such a way that the condition is satisfied.

Let’s assume the positions of **P** and **S** are determined by choosing the position of **P** and placing **S** 5 positions apart (so that there are 4 letters between them). 

Since the word has 12 letters, the possible positions for **P** are as follows:

- If **P** is in position 1, **S** must be in position 6.
- If **P** is in position 2, **S** must be in position 7.
- If **P** is in position 3, **S** must be in position 8.
- If **P** is in position 4, **S** must be in position 9.
- If **P** is in position 5, **S** must be in position 10.
- If **P** is in position 6, **S** must be in position 11.
- If **P** is in position 7, **S** must be in position 12.
  
Alternatively, **P** could be placed after **S** in the same way. For each of the above cases, we could swap **P** and **S**. Thus, there are \(7 \times 2 = 14\) ways to place **P** and **S** with 4 letters between them.

### Step 3: Arrange the remaining 10 letters
Now that **P** and **S** are fixed, we are left with 10 other positions to arrange the remaining 10 letters: E, R, M, U, T, T, A, I, O, N.

Since the letter **T** is repeated twice, the number of distinct ways to arrange these 10 letters is given by:
\[
\frac{10!}{2!}
\]
where \(10!\) is the total number of ways to arrange 10 letters, and \(2!\) accounts for the repeated **T**.

### Step 4: Calculate the total number of arrangements
Thus, the total number of arrangements is:
\[
14 \times \frac{10!}{2!}
\]
Now, let's calculate the value of this expression.

\[
10! = 3,628,800
\]
\[
2! = 2
\]
\[
\frac{10!}{2!} = \frac{3,628,800}{2} = 1,814,400
\]
Finally, multiply by 14:
\[
14 \times 1,814,400 = 25,401,600
\]

### Final Answer:
There are **25,401,600** ways to arrange the letters of the word "PERMUTATIONS" such that there are always 4 letters between **P** and **S**.

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions to expand your understanding:
1. How many ways can the letters of "PERMUTATIONS" be arranged with no restrictions?
2. How many ways can the letters of "PERMUTATIONS" be arranged if the letters **T** must always be next to each other?
3. How many distinct arrangements are there for the word "STATISTICS"?
4. What if there must be exactly 3 letters between **P** and **S**? How does this affect the number of arrangements?
5. In how many ways can the letters of "PERMUTATIONS" be arranged if **P**, **E**, and **R** must always appear together?

Tip: When dealing with repeated letters, always divide by the factorial of the number of times each repeated letter appears to avoid overcounting.

In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

Explore how to calculate the number of ways to arrange the letters of 'PERMUTATIONS' such that there are always 4 letters between P and S. Learn key concepts in combinatorics, including permutations and factorials, with a step-by-step breakdown. Suitable for Grades 9-12.

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