A website requires users to create a password that is 6 characters long. Each character can contain numbers, or symbols chosen from ? and #. The password does not have to include both numbers and symbols. a) How many 6-character passwords are possible if numbers and symbols can be used more than once? b) How many 6-character passwords are possible if no numbers or symbols can be repeated? c) What is the probability that a randomly generated 6-character password contains 6 different characters? a) Set up the expression that can be used to calculate the number of possible 6-character passwords if numbers and symbols can be used more than once. The expression is (Do not simplify.) The number of different passwords that are possible if numbers and symbols can be used more than once is (Simplify your answer.) b) The number of different passwords that are possible if no numbers or symbols can be repeated is (Simplify your answer.) c) The probability that a 6-character password contains 6 different characters is (Type an integer or decimal rounded to three decimal places as needed.)
The Correct Answer and Explanation is:
Let’s go step by step for each part of the question:
Part (a) – Number of 6-character passwords where numbers and symbols can be repeated:
There are 10 possible digits (0-9) and 2 symbols (? and #), so a total of 12 possible characters (10 numbers + 2 symbols) can be chosen for each character in the password.
Since the password is 6 characters long and characters can be repeated, the total number of different 6-character passwords can be calculated as:12×12×12×12×12×12=12612 \times 12 \times 12 \times 12 \times 12 \times 12 = 12^612×12×12×12×12×12=126
So the expression for the number of possible passwords is:12612^6126
Now, let’s calculate the value:126=2,985,98412^6 = 2,985,984126=2,985,984
Thus, 2,985,984 different passwords are possible if numbers and symbols can be repeated.
Part (b) – Number of 6-character passwords with no repetition of numbers or symbols:
For passwords where no numbers or symbols can be repeated, we must choose 6 different characters from the 12 possible characters (10 digits + 2 symbols). The first character has 12 options, the second character has 11 options (since one character has already been used), and so on.
Thus, the total number of possible passwords is:12×11×10×9×8×7=12!/(12−6)!12 \times 11 \times 10 \times 9 \times 8 \times 7 = 12! / (12-6)!12×11×10×9×8×7=12!/(12−6)!
Calculating the product:12×11×10×9×8×7=665,28012 \times 11 \times 10 \times 9 \times 8 \times 7 = 665,28012×11×10×9×8×7=665,280
Thus, 665,280 different passwords are possible if no characters are repeated.
Part (c) – Probability that a randomly generated 6-character password contains 6 different characters:
The probability that a randomly generated password has 6 different characters is the ratio of the number of passwords with no repeated characters to the total number of passwords with repeated characters. From parts (a) and (b), we have:
- Total number of passwords (with repetitions allowed) = 2,985,984
- Number of passwords with no repetitions = 665,280
So, the probability is:665,2802,985,984≈0.223\frac{665,280}{2,985,984} \approx 0.2232,985,984665,280≈0.223
Thus, the probability that a 6-character password contains 6 different characters is approximately 0.223 or 22.3%.
Summary of answers:
a) The expression for the number of passwords is 12612^6126, and the total number of passwords is 2,985,984.
b) The total number of passwords with no repetition is 665,280.
c) The probability of a password containing 6 different characters is approximately 0.223.
