Sure, let's walk through the problem step by step.
### Without Reusing Characters
1. Using Letters Without Reuse:
- 26 options for the first letter.
- 25 options for the second, since one letter has been used.
- 24 options for the third, since two letters have been used.
- 23 options for the fourth, since three letters have been used.
- 22 options for the fifth, since four letters have been used.
- 21 options for the sixth, since five letters have been used.
So, the calculation for the six letters without reuse looks like this:
[tex]$ 26 \times 25 \times 24 \times 23 \times 22 \times 21 $[/tex]
2. Using Numbers Without Reuse:
- 10 options for the first number.
- 9 options for the second number, since one number has been used.
So, the calculation for the two numbers without reuse looks like this:
[tex]$ 10 \times 9 $[/tex]
3. Combining Both:
Combining the calculations for letters and numbers without reuse:
[tex]$ 26 \times 25 \times 24 \times 23 \times 22 \times 21 \times 10 \times 9 = 14,918,904,000 $[/tex]
So, the total number of unique passwords without reuse is 14,918,904,000.
### With Reusing Characters
1. Using Letters With Reuse:
- For each position, you have all 26 letters available, as characters can be reused.
Thus, for six letters:
[tex]$ 26 \times 26 \times 26 \times 26 \times 26 \times 26 = 26^6 $[/tex]
2. Using Numbers With Reuse:
- For each position, you have all 10 numbers available, as numbers can be reused.
Thus, for two numbers:
[tex]$ 10 \times 10 = 10^2 $[/tex]
3. Combining Both:
Combining the calculations for letters and numbers with reuse:
[tex]$ 26^6 \times 10^2 $[/tex]
Which amounts to:
[tex]$ 26^6 \times 10^2 = 30,891,577,600 $[/tex]
So, the total number of possible passwords with reuse is 30,891,577,600.
### Summary
1. Number of unique passwords without reusing any character: 14,918,904,000.
2. Number of possible passwords with reuse of characters: 30,891,577,600.
That’s almost 30.9 billion passwords!
