Q: Count the number of passwords with following constraints. Assuming digits = {0, 1, …..9},
letters = {a, b, ….. y, z}
(a) 5 characters which are digits or letters.
(b) 4 characters which are digits or letters, with at least 1 digit.
(c) 4 characters which are digits. Adjacent characters are not the same.
(d) 6 characters which are letters. The first half must not equal to the last half.
(a) 36^5
(b) 10*36^4
(c) 10*9*9*9
(d) assuming you just mean that the exact sequence of 3 letters does not repeat, then 36P3 * (36P3 - 1)
extra credit: what if you mean that the first 3 letters do not all repeat in the 2nd half of the password?
36P3 * (36-1 * 35-1 * 34-1)
is it correct.
A: Sure, I'd be happy to help! Let's break it down one by one:
(a) For a 5 character password with digits or letters, we have 36 options for the first character (26 letters + 10 digits), 36 options for the second character, and so on. Therefore, the total number of passwords would be 36^5.
(b) For a 4 character password with digits or letters, with at least 1 digit, we have 36 options for the first character (26 letters + 10 digits), 36 options for the second character, and 36 options for the third character. For the fourth character, since it must be a digit, we have 10 options. So the total number of passwords would be 36^3 * 10.
(c) For a 4 character password with digits, where adjacent characters are not the same, we have 10 options for the first character (digits 0-9), 9 options for the second character (since it can't be the same as the first), 9 options for the third character, and 9 options for the fourth character. Therefore, the total number of passwords would be 10 * 9^3.
(d) For a 6 character password with letters, where the first half must not equal the last half, we have 26 options for the first character, 26 options for the second character, and so on, for a total of 26^6 possible passwords.
I hope that answers your question!
(a) To find the number of passwords with 5 characters which are digits or letters, we need to calculate the total number of possible combinations.
Since we have 26 letters (a-z) and 10 digits (0-9), the total number of characters is 26 + 10 = 36.
Therefore, the number of passwords can be calculated as:
36 * 36 * 36 * 36 * 36 = 36^5 = 60,466,176
So, there are 60,466,176 possible passwords with 5 characters which are digits or letters.
(b) To find the number of passwords with 4 characters which are digits or letters, with at least 1 digit, we need to calculate the total number of combinations considering the different possibilities for the digit placement.
Since we have 26 letters and 10 digits, the total number of characters is still 36.
To have at least 1 digit in a 4-character password, we can have 3 letters and 1 digit or 2 letters and 2 digits.
The number of combinations for 3 letters and 1 digit is: 36 * 36 * 36 * 10 = 46,656
The number of combinations for 2 letters and 2 digits is: 36 * 36 * 10 * 10 = 12,960
Therefore, the total number of passwords is: 46,656 + 12,960 = 59,616
So, there are 59,616 possible passwords with 4 characters which are digits or letters, with at least 1 digit.
(c) To find the number of passwords with 4 characters which are digits, where adjacent characters are not the same, we need to calculate the combinations while ensuring that no two adjacent characters are the same.
Since we have 10 digits (0-9), we can choose any digit for the first character (10 options). For the second character, we can choose any digit except the one chosen for the first character (9 options). Similarly, for the third character, we have 9 options, and for the fourth character, we have 9 options.
Therefore, the total number of passwords is: 10 * 9 * 9 * 9 = 7,290
So, there are 7,290 possible passwords with 4 characters which are digits, and adjacent characters are not the same.
(d) To find the number of passwords with 6 characters which are letters, where the first half must not equal the last half, we can calculate the combinations by considering the different possibilities:
Since we have 26 letters, we can choose any letter for the first character (26 options). For the second character, we can choose any letter except the one chosen for the first character (25 options). Similarly, for the third character, we have 25 options, and for the fourth character, we have 25 options.
Therefore, the total number of passwords is: 26 * 25 * 25 * 25 * 25 * 25 = 6,890,625
So, there are 6,890,625 possible passwords with 6 characters which are letters, and the first half does not equal the last half.
(a) To count the number of passwords with 5 characters which are digits or letters, you need to consider the total number of possible characters and raise it to the power of 5.
The total number of possible characters is the sum of the number of digits and the number of letters. Assuming there are 10 digits (0-9) and 26 letters (a-z), the total number of possible characters is 10 + 26 = 36.
Therefore, the total number of passwords with 5 characters can be calculated as 36^5.
(b) To count the number of passwords with 4 characters which are digits or letters with at least 1 digit, we can divide the problem into two parts:
(i) The number of passwords with 4 characters which are digits or letters (no restriction on digits)
(ii) The number of passwords with 4 characters which are letters only (no digits allowed)
In the first part, we have the same calculation as in part (a), which is 36^4.
In the second part, we need to count the number of passwords with 4 characters formed only by letters. Since there are 26 letters available, the number of these passwords is 26^4.
Therefore, the total number of passwords with 4 characters and at least 1 digit can be calculated as (36^4) - (26^4).
(c) To count the number of passwords with 4 characters which are digits, where adjacent characters are not the same, we can approach it using simple combinatorics.
For the first digit, we have 10 choices (0-9). For the second digit, since it cannot be the same as the first one, we have 9 choices. The same goes for the third digit, with 9 choices. Finally, for the fourth digit, we again have 9 choices.
Therefore, the total number of passwords with 4 digits and non-adjacent characters is 10 * 9 * 9 * 9.
(d) To count the number of passwords with 6 characters which are letters, where the first half must not equal the last half, we can approach it as follows:
For the first three characters (the first half), we have 26 choices for each character, giving us 26^3 possibilities.
For the last three characters (the last half), again we have 26 choices for each character, giving us another 26^3 possibilities.
Therefore, the total number of passwords with 6 characters and the first half not equal to the last half is (26^3) * (26^3).