Question

What is the total number of different nine-letter arrangements that can be formed using the letters in the word “TENNESSEE”?(1 point)
Responses

3,780
3,780

15,120
15,120

45,360
45,360

362,880

Answer 1

To find the total number of different nine-letter arrangements that can be formed using the letters in the word "TENNESSEE", we first note the frequency of each letter:

- T: 1
- E: 4
- N: 2
- S: 2

The total number of arrangements of a word with letters that may repeat can be calculated using the formula for permutations of a multiset:

\[
\text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdot ...}
\]

where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3, ... \) are the frequencies of the repeated letters.

For "TENNESSEE", we have:

- \( n = 9 \) (total letters)
- For T, E, N, and S:
- \( n_T = 1! \) (T appears once)
- \( n_E = 4! \) (E appears four times)
- \( n_N = 2! \) (N appears twice)
- \( n_S = 2! \) (S appears twice)

Now we can plug in these values to calculate:

\[
\text{Number of arrangements} = \frac{9!}{1! \cdot 4! \cdot 2! \cdot 2!}
\]

Calculating the factorials:

- \( 9! = 362,880 \)
- \( 1! = 1 \)
- \( 4! = 24 \)
- \( 2! = 2 \)

Now substituting these into the formula:

\[
\text{Number of arrangements} = \frac{362,880}{1 \cdot 24 \cdot 2 \cdot 2}
\]

Calculating the denominator:

\[
1 \cdot 24 \cdot 2 \cdot 2 = 96
\]

Now we compute:

\[
\text{Number of arrangements} = \frac{362,880}{96} = 3,780
\]

Thus, the total number of different nine-letter arrangements that can be formed using the letters in "TENNESSEE" is **3,780**.