There are several ways to arrange the letters CORRESPONDENTS so that there are only three consecutive pairs of identical letters.

We are tasked with finding how many ways we can arrange the letters in the word **CORRESPONDENTS** such that there are exactly three consecutive pairs of identical letters.

### Step 1: Analyzing the structure of the word

The word **CORRESPONDENTS** consists of the following letters:

- C: 1
- O: 2
- R: 2
- E: 2
- S: 2
- P: 1
- N: 1
- D: 1
- T: 1

In total, the word has 13 letters, with several repeated ones.

### Step 2: Identifying pairs of identical letters

Pairs of identical letters refer to two adjacent letters that are the same, such as **RR** or **SS**. The letters that can potentially form consecutive pairs are:

- O (2 occurrences)
- R (2 occurrences)
- E (2 occurrences)
- S (2 occurrences)

We need to form exactly **three consecutive pairs** from these.

### Step 3: Selecting which letters will form the pairs

Since we have 4 types of letters that can potentially form pairs (O, R, E, S), we need to choose 3 of these to form pairs. The number of ways to select 3 letters out of these 4 is given by the combination formula \( \binom{4}{3} \), which evaluates to:

\[
\binom{4}{3} = 4
\]

### Step 4: Arranging the remaining letters

Once we have fixed the three pairs, we now have to arrange the remaining letters. The total number of letters is 13, and three of these are involved in pairs, which means they count as 6 letters in total. This leaves us with \( 13 - 3 = 10 \) letters to arrange.

Of these remaining letters, we have:

- C: 1
- P: 1
- N: 1
- D: 1
- T: 1
- The single letters from the repeated letters (whichever was not paired)

Now we need to determine the total number of ways to arrange these remaining 10 letters.

### Step 5: Calculating the total number of arrangements

This step involves calculating the number of permutations of the letters, taking into account the repeated letters.

For further accuracy, additional steps are required.

Explore how to arrange the letters of CORRESPONDENTS such that there are exactly three consecutive pairs of identical letters. This problem involves combinatorics concepts like permutations and combinations, perfect for students in Grades 10-12.

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