Calculating 4-Character Password Possibilities
The problem asks for the number of distinct 4-character passwords that can be formed using digits $0 - 9$ and letters $A - F$ without repetition.
Character Set Identification
- Digits available: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ (10 digits)
- Letters available: $A, B, C, D, E, F$ (6 letters)
- Total available unique characters: $10 + 6 = 16$
- Password length required: 4 characters
- Constraint: No character repetition allowed
Permutation Calculation
Since the order of characters matters in a password and repetition is not allowed, this is a permutation problem. We need to find the number of permutations of selecting 4 characters from a set of 16 distinct characters.
The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$, where $n$ is the total number of items to choose from, and $k$ is the number of items to choose.
In this case, $n = 16$ and $k = 4$.
Number of possible passwords = $P(16, 4)$
$ P(16, 4) = \frac{16!}{(16-4)!} = \frac{16!}{12!} $ $ P(16, 4) = 16 \times 15 \times 14 \times 13 $Result
Performing the multiplication:
- $16 \times 15 = 240$
- $14 \times 13 = 182$
- $240 \times 182 = 43680$
Therefore, there are 43,680 possible 4-character passwords.
Important Questions from Permutation and Combination (Notes)
- In how many ways can you place $N$ coins on a board with $N$ rows and $N$ columns such that every row and every column contains exactly one coin?
- The number of three English letter words, having at least one consonant, but not having two consecutive consonants, is
- The number of three digit PINs, in which the third digit is the sum of the first two digits, is
- How many four-digit ATM PINs which read the same from left to right and right to left are possible?
- There are 8 different symbols. A code is formed by selecting 5 symbols such that each symbol can be used only once (that is, no symbol is repeated in the code). Also, the order of symbols matters. How many different codes can be formed ?
