The Password Problem

What do you think?

A password has 4 characters: one uppercase, one lowercase, then two digits. How many possible passwords exist?

You could start listing them: Aa00, Aa01, Aa02...

You'd be at it for a while. There's a faster way.

Start smaller

What do you think?

You have 3 shirts and 2 pants. How many different outfits can you make?

Enter a whole number

Outfit Grid

Wardrobe Builder

shirts (3)

pants (2)

3×2=6combinations

all combinations

Tap the items on and off. Watch the total change.

With 3 shirts and 2 pants, you get 6 outfits. Not because you listed them, but because each shirt pairs with every pair of pants.

With 4 shirts and 3 pants, how many outfits? (whole number)

Now add 2 hats. With 4 shirts, 3 pants, 2 hats, how many complete outfits? (whole number)

The rule

The Multiplication Rule

If you have $n_1$ ways to make the first choice, $n_2$ ways to make the second, and so on, then the total number of ways is: $n_1 \times n_2 \times \cdots \times n_k$

This works because the choices are independent. Picking the purple shirt doesn't eliminate any pants.

Back to passwords

Counting the passwords

\text{Uppercase: } 26 \text{ choices}

A-Z gives us 26 options for the first character.

Step 1 of 5

Password strength explorer

Length: 4

1816

character types

uppercase

+26

ABCDEFGHIJKLMNOPQRSTUVWXYZ

lowercase

+26

abcdefghijklmnopqrstuvwxyz

digits

+10

0123456789

symbols

+32

!@#$%^&*()_+-=[]{}|;:,.<>?

alphabet^length

62⁴ = 14,776,336

time to crack @ 1B/sec

weak15ms

Using 1 more character adds 62 combinations

A 6-digit PIN (0-9 for each digit). How many possible PINs? (whole number)

A 4-letter password (lowercase only). How many possibilities? (whole number)

With vs. without replacement

What do you think?

A 4-digit PIN where digits CAN'T repeat. First digit has 10 options. How many options for the second digit?

Enter a whole number

PIN Code Builder

position 1: 10 choices

total 4-digit PINs

10 × 10 × 10 × 10 = 10^4

10,000

Type	How it works	4-digit PIN count
With replacement	10, 10, 10, 10	10,000
Without replacement	10, 9, 8, 7	5,040

With replacement: every position has the same choices.

Without replacement: the numbers shrink: $n$ , then $n-1$ , then $n-2$ ...

3-digit PIN, no repeating digits. How many options? (whole number)

3-letter code (A-Z), no repeating letters. How many codes? (whole number)

Why this matters for security

What do you think?

Standard 4-digit PIN (10,000 possibilities). At 1000 guesses per second, how long to try them all?

Password Type	Possibilities	Time to Crack @ 1B/sec
4-digit PIN	10,000	Instant
6-char (a-z)	309 million	< 1 second
8-char (a-z, A-Z, 0-9)	218 trillion	~2.5 days
12-char (a-z, A-Z, 0-9)	3 × 10²¹	~96,000 years

Each additional character multiplies the count by the alphabet size, which is why longer passwords are exponentially more secure.

The shrinking pattern

When choices shrink at each step: $n \times (n-1) \times (n-2) \times \cdots$

This pattern comes up so often that mathematicians gave it a name and a symbol. This is covered in the combinations vs permutations lesson.

5 people in a race. How many ways can they finish 1st, 2nd, 3rd? (No ties) (whole number)

License plate: 3 letters then 4 digits. Repetition allowed. How many plates? (whole number)

True or False: Adding one character to a password roughly DOUBLES the possibilities. (true or false)

The next lesson covers factorials, permutations, and combinations.