Permutations and Factorials

Permutations

• Permutations are the number of arrangements possible when order matters.
• Example: Passwords, where the order of characters is crucial.

Definition of Permutation

• Represented as $nPr$, where:
  • $P$ is the number of permutations we're solving for.
  • $n$ is the number of distinct objects.
  • $r$ is the number of objects taken at a time.

Example: Phone Passcode

• Consider a phone passcode using digits 0-9.
• If each digit can only be used once (distinct objects), we can use the permutation formula.

Factorials

• The exclamation mark (!) in math denotes a factorial.
• $n! = n \times (n-1) \times (n-2) \times … \times 1$
• Example: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Using Permutations in the Real World

• Scenario: you're trying to guess a friend's four-digit phone passcode.
• Assume you observe your friend using unique numbers (no repetition) when unlocking the phone.

Applying the Permutation Formula

• $n = 10$ (since there are 10 unique digits: 0-9).
• $r = 4$ (since the passcode is four digits long).
• The permutation formula is: $nPr = \frac{n!}{(n-r)!}$

Calculation

$\frac{10!}{(10-4)!} = \frac{10!}{6!}$

• Expanding the factorials:

$\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$

• Simplifying by canceling out common terms (6!):

$\frac{10 \times 9 \times 8 \times 7 \times 6!}{6!} = 10 \times 9 \times 8 \times 7$

Computing the Result

• $9 \times 8 \times 7 = 504$
• $504 \times 10 = 5040$
• Therefore, $10P4 = 5040$ possible permutations.

Implications

• There are 5,040 different possible combinations for a four-digit passcode with unique digits.
• Phones have security measures (like shutting down or erasing data after multiple incorrect attempts) to prevent unauthorized access via automated permutation attempts.