If the letters of the word REGULATIONS be arranged at random, find the probability that there will be exactly four letters between the `R` and the`Edot`

Explanation

To solve the problem of finding the probability that there will be exactly four letters between the letters 'R' and 'E' in the word "REGULATIONS", we can follow these steps:

### Step 1: Determine the total number of arrangements of the letters in "REGULATIONS".

The word "REGULATIONS" consists of 11 letters. The total number of arrangements of these letters is given by:

\[ \text{Total arrangements} = 11! \]

### Step 2: Identify the requirement for the arrangement.

We need to find the arrangements where there are exactly four letters between 'R' and 'E'. This means we can visualize the arrangement as follows:

- R _ _ _ _ E (or E _ _ _ _ R)

Here, the underscores represent the four letters that will be placed between 'R' and 'E'.

### Step 3: Choose the letters to fill the gaps.

Since 'R' and 'E' are already fixed, we have 9 remaining letters from which we need to choose 4 letters to fill the gaps. The number of ways to choose 4 letters from the remaining 9 letters is given by:

\[ \text{Ways to choose 4 letters} = \binom{9}{4} \]

### Step 4: Arrange the chosen letters.

The 4 letters chosen can be arranged among themselves in:

\[ 4! \text{ ways} \]

### Step 5: Arrange 'R' and 'E'.

The letters 'R' and 'E' can be arranged in 2 different ways (either 'R' first or 'E' first):

\[ 2! \text{ ways} \]

### Step 6: Arrange the remaining letters.

After placing 'R', 'E', and the 4 chosen letters, there are 6 letters left (since we started with 11 letters and used 5). These remaining letters can be arranged in:

\[ 6! \text{ ways} \]

### Step 7: Calculate the total favorable arrangements.

Now, we can calculate the total number of favorable arrangements where 'R' and 'E' have exactly four letters between them:

\[ \text{Favorable arrangements} = \binom{9}{4} \times 4! \times 2! \times 6! \]

### Step 8: Calculate the probability.

The probability \( P \) that there are exactly four letters between 'R' and 'E' is given by the ratio of favorable arrangements to total arrangements:

\[ P = \frac{\binom{9}{4} \times 4! \times 2! \times 6!}{11!} \]

### Step 9: Simplify the expression.

Now we can simplify this expression step by step:

1. Calculate \( \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!} \).
2. Substitute this into the probability formula.
3. Calculate \( 11! = 11 \times 10 \times 9! \).
4. Cancel \( 9! \) in the numerator and denominator.
5. Simplify the remaining factorials.

### Final Expression:

After performing the calculations, you will arrive at the final probability.

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