Question
How many 6-letter passwords are possible if letters cannot be used more than once?
Answers
Answered by
GPT-4o mini
To find the total number of possible 6-letter passwords using the English alphabet (which consists of 26 letters) without repetition, we can use the principle of counting permutations.
In a password of 6 letters:
1. We can choose the first letter in 26 different ways (since there are 26 letters in total).
2. After choosing the first letter, we have 25 letters left for the second position.
3. For the third position, we will have 24 letters remaining (after choosing the first two letters).
4. For the fourth position, we will have 23 letters left.
5. For the fifth position, there will be 22 letters available.
6. Finally, for the sixth position, we will have 21 letters to choose from.
To calculate the total number of different 6-letter passwords, we multiply the number of choices for each position together:
\[
26 \times 25 \times 24 \times 23 \times 22 \times 21
\]
Now, let's calculate that step by step:
- First, calculate \( 26 \times 25 = 650 \)
- Then, \( 650 \times 24 = 15600 \)
- Next, \( 15600 \times 23 = 358800 \)
- Continuing, \( 358800 \times 22 = 7901760 \)
- Finally, \( 7901760 \times 21 = 165772160 \)
Thus, the total number of possible 6-letter passwords, where no letters are repeated, is
\[
\boxed{165772160}
\]
In a password of 6 letters:
1. We can choose the first letter in 26 different ways (since there are 26 letters in total).
2. After choosing the first letter, we have 25 letters left for the second position.
3. For the third position, we will have 24 letters remaining (after choosing the first two letters).
4. For the fourth position, we will have 23 letters left.
5. For the fifth position, there will be 22 letters available.
6. Finally, for the sixth position, we will have 21 letters to choose from.
To calculate the total number of different 6-letter passwords, we multiply the number of choices for each position together:
\[
26 \times 25 \times 24 \times 23 \times 22 \times 21
\]
Now, let's calculate that step by step:
- First, calculate \( 26 \times 25 = 650 \)
- Then, \( 650 \times 24 = 15600 \)
- Next, \( 15600 \times 23 = 358800 \)
- Continuing, \( 358800 \times 22 = 7901760 \)
- Finally, \( 7901760 \times 21 = 165772160 \)
Thus, the total number of possible 6-letter passwords, where no letters are repeated, is
\[
\boxed{165772160}
\]