How many different words can be formed with the letters of the word REGURGITATE so that the two T's are always together?

Options

A. a. 453600
B. b. 543600
C. c. 18360
D. d. 5040

Answer

To solve the problem of how many different words can be formed with the letters of the word "REGURGITATE" such that the two T's are always together, we can follow these steps:

### Step 1: Treat the two T's as a single unit
Since the two T's must always be together, we can treat them as one single unit. Let's denote this unit as "TT".

### Step 2: Count the total letters
Now, instead of counting 11 letters (R, E, G, U, R, G, I, T, A, T, E), we will count the letters as follows:
- R
- E
- G
- U
- R
- G
- I
- A
- TT (the combined unit of T's)

This gives us a total of 10 units: R, E, G, U, R, G, I, A, and TT.

### Step 3: Identify the frequency of each letter
Next, we need to identify how many times each letter appears:
- R appears 2 times
- E appears 1 time
- G appears 2 times
- U appears 1 time
- I appears 1 time
- A appears 1 time
- TT appears 1 time

### Step 4: Use the formula for permutations of multiset
The formula for the number of permutations of a multiset is given by:

\[ \text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]

Where:
- \( n \) is the total number of items,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the distinct items.

In our case:
- Total letters (n) = 10
- Frequencies: R = 2, G = 2, E = 1, U = 1, I = 1, A = 1, TT = 1

So we can plug in the values:

\[ \text{Number of permutations} = \frac{10!}{2! \times 2! \times 1! \times 1! \times 1! \times 1!} \]

### Step 5: Calculate the factorials
Now we can compute this:

\[ 10! = 3628800 \]
\[ 2! = 2 \]

Thus, we have:

\[ \text{Number of permutations} = \frac{3628800}{2 \times 2 \times 1 \times 1 \times 1 \times 1} = \frac{3628800}{4} = 907200 \]

### Final Answer
The total number of different words that can be formed with the letters of the word "REGURGITATE" such that the two T's are always together is **907200**.

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