To determine how many different permutations can be formed using all the letters in the word "COLORADO," we can follow these steps:
1. Determine the total number of letters: The word "COLORADO" contains 8 letters.
2. Identify the frequency of each distinct letter:
- C appears 1 time,
- O appears 3 times,
- L appears 1 time,
- R appears 1 time,
- A appears 1 time,
- D appears 1 time.
3. Calculate the factorial of the total number of letters: The factorial of 8 (denoted as [tex]\(8!\)[/tex]) represents the total number of arrangements of 8 distinct letters.
[tex]\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\][/tex]
4. Divide by the factorial of the frequency of each repeating letter: We do this to account for the repeated letters, as these repetitions reduce the number of unique permutations.
- For the letter O, which appears 3 times, we divide by [tex]\(3!\)[/tex]:
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
5. Calculate the number of unique permutations:
[tex]\[
\text{Number of permutations} = \frac{8!}{3!} = \frac{40320}{6} = 6720
\][/tex]
Thus, the number of different permutations that can be formed using all the letters in the word "COLORADO" is 6720.
