To find the total number of different nine-letter arrangements that can be formed using the letters in the word "TENNESSEE," we first identify the letters and their frequencies:
- T: 1
- E: 4
- N: 2
- S: 2
The formula for the number of arrangements of letters where some letters are repeated is given by:
\[ \frac{n!}{n_1! \times n_2! \times n_3! \times ...} \]
where \( n \) is the total number of letters, and \( n_1, n_2, ... \) are the frequencies of the repeated letters.
In the case of "TENNESSEE":
- Total letters, \( n = 9 \) (T, E, N, N, E, S, S, E, E)
- Frequencies: T = 1, E = 4, N = 2, S = 2
Using the formula:
\[ \text{Arrangements} = \frac{9!}{1! \times 4! \times 2! \times 2!} \]
First, we calculate \( 9! \):
\[ 9! = 362,880 \]
Next, we calculate the factorials for the repeated letters:
- \( 1! = 1 \)
- \( 4! = 24 \)
- \( 2! = 2 \)
- \( 2! = 2 \)
Now we substitute these values into our formula:
\[ \text{Arrangements} = \frac{362,880}{1 \times 24 \times 2 \times 2} \]
Calculating the denominator:
\[ 1 \times 24 \times 2 \times 2 = 96 \]
Now we compute the final number of arrangements:
\[ \text{Arrangements} = \frac{362,880}{96} = 3,780 \]
Thus, the total number of different nine-letter arrangements that can be formed using the letters in the word "TENNESSEE" is 3,780.