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 * 5
see what you can do with (c) and (d)
(a) To calculate the number of passwords with 5 characters which are either digits or letters, we can calculate the total number of possible combinations.
There are 26 letters in the alphabet (a to z) and 10 digits (0 to 9), which gives us a total of 36 characters to choose from.
Each character in the password has 36 choices, and since there are 5 characters in total, the number of possible passwords can be calculated as:
Total number of passwords = 36^5
(b) To calculate the number of passwords with 4 characters which are either digits or letters, with at least 1 digit, we need to consider two cases.
Case 1: One digit and three characters which are letters.
In this case, we have 10 choices for the digit and 26 choices for each of the three letters. Hence the total number of passwords in this case can be calculated as:
Number of passwords = 10 * 26^3
Case 2: Four characters which can be either letters or digits.
In this case, we have 36 choices for each of the four characters. Hence the total number of passwords in this case can be calculated as:
Number of passwords = 36^4
To get the total number of passwords with at least 1 digit, we add the number of passwords from both cases:
Total number of passwords = 10 * 26^3 + 36^4
(c) To calculate the number of passwords with 4 digits where adjacent characters are not the same, we have the following options:
For the first digit, we have 10 choices (0-9).
For each of the next three digits, we have 9 choices (excluding the digit chosen for the previous one).
Hence, the total number of passwords can be calculated as:
Total number of passwords = 10 * 9 * 9 * 9 = 7290
(d) To calculate the number of passwords with 6 letters where the first half does not equal the last half, we need to consider the following cases:
Case 1: Three distinct letters in the first half and three distinct letters in the second half.
In this case, we have 26 choices for each of the letters in the first half and 23 choices for each of the letters in the second half (excluding the three letters chosen for the first half).
Total number of passwords in this case = 26 * 25 * 24 * 23 * 22 * 21
Case 2: Two distinct letters repeated twice in the first half and two distinct letters repeated twice in the second half.
In this case, we have 26 choices for the letters in the first half (excluding two distinct letters) and 23 choices for the letters in the second half (excluding the two distinct letters).
Total number of passwords in this case = (26 * 25/2) * (23 * 22/2) * (24 * 23/2) * (21 * 20/2)
To get the total number of passwords, we add the number of passwords from both cases:
Total number of passwords = 26 * 25 * 24 * 23 * 22 * 21 + (26 * 25/2) * (23 * 22/2) * (24 * 23/2) * (21 * 20/2)
To count the number of passwords with the given constraints, let's analyze each case separately:
(a) 5 characters which are digits or letters:
In this case, we have a total of 36 possible characters (26 letters + 10 digits). Since each character can be chosen independently, the total number of passwords is given by 36^5.
(b) 4 characters which are digits or letters, with at least 1 digit:
To calculate the number of passwords in this case, we can split it into two parts:
1) The first character is a digit, and the remaining 3 characters can be any letter or digit. The number of passwords in this case is 10 * 36^3.
2) The first character is a letter, and the remaining 3 characters can be any letter or digit. The number of passwords in this case is 26 * 36^3.
The total number of passwords in case (b) is obtained by adding the results of the above two parts.
(c) 4 characters which are digits. Adjacent characters are not the same:
In this case, there are 10 choices for the first digit. For each subsequent digit, we need to ensure that it is different from the previous one. Hence, we have 9 choices for the second digit, 9 choices for the third digit, and 9 choices for the fourth digit. Therefore, the total number of passwords in this case is 10 * 9^3.
(d) 6 characters which are letters. The first half must not equal to the last half:
For the first half, we have 26 choices for each of the 3 characters. For the second half, we need to ensure that each character is different from its corresponding character in the first half (e.g., the first character of the second half should not be the same as the first character of the first half). Hence, we have 25 choices for each of the 3 characters in the second half. Therefore, the total number of passwords in this case is 26^3 * 25^3.
To summarize:
(a) Number of passwords = 36^5
(b) Number of passwords = (10 * 36^3) + (26 * 36^3)
(c) Number of passwords = 10 * 9^3
(d) Number of passwords = 26^3 * 25^3