Does order matter?

What do you think?

You're picking 3 people from a group of 5. In one case you're electing President, VP, and Secretary. In another, you're just forming a committee. Which has more possible outcomes?

When order matters: permutations

You're assigning ranked positions. Tap people below to fill the roles.

P(5,3) → C(5,3)

Ranked Selection

Available (5)

Ranked Selection (0/3)

1st5 choices

2nd4 choices

3rd3 choices

Permutations P(5,3)

5 × 4 × 3

What do you think?

You picked a President (5 choices). How many choices remain for VP?

Enter a whole number

This gives us: $5 \times 4 \times 3 = 60$ arrangements.

The "shrinking choices" pattern — $5 \times 4 \times 3$ — shows up so often it gets its own notation.

Factorial

$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$

By convention, $0! = 1$ (there is exactly one way to arrange zero objects: do nothing).

Examples: $5! = 120$ , $3! = 6$ , $1! = 1$ , $0! = 1$ .

Permutation

An arrangement where order matters. $P(n,k) = \Large\frac{n!}{(n-k)!} \normalsize= n \times (n-1) \times \cdots \times (n-k+1)$

How many ways can you arrange 4 books on a shelf from a collection of 7? (whole number)

A race has 10 runners. How many ways can gold, silver, and bronze be awarded? (whole number)

When order doesn't matter: combinations

Now you're just picking a committee. No roles, just a group.

Pick a Committee

Your Committee (0/3)

Tap items above to add them

C(5,3)

P(5,3)

60 arrangements ÷ 3! orderings

60 ÷ 6 = 10

Watch what happens. You tap A, B, C, but the result is just {A, B, C}. The order you tapped doesn't create a different committee.

What do you think?

We counted 60 permutations for officer positions. For any group of 3 items (like A, B, C), how many ways can they be arranged?

Combination

A selection where order doesn't matter. $\binom{n}{k} = \Large\frac{n!}{k!(n-k)!}$

How many ways can you choose 3 toppings from 8 options for a pizza? (whole number)

A club has 12 members. How many 4-person committees can be formed? (whole number)

The relationship

Permutations count every ordering. Combinations collapse them.

From Permutations to Combinations

P(n,k) = \frac{n!}{(n-k)!}

Start with the permutation formula, counts all ordered arrangements.

Step 1 of 3

See the overcounting in action. Watch all 60 permutations of choosing 3 from {A,B,C,D,E} collapse into 10 combinations:

Why Divide by k!

Choose 3 from {A, B, C, D, E}. Click any arrangement to see its duplicates.

Permutations

P(5,3) = 60

÷ orderings

3! = 6

Combinations

C(5,3) = 10

Poker hands

A poker hand is 5 cards from a 52-card deck.

What do you think?

When you're dealt a poker hand, does the order you receive the cards matter?

Poker Hand Builder

Your Hand (0/5)

Tap cards above to build your hand

C(52,5)

2,598,960

Ways to complete

How many 5-card poker hands are possible? (Enter the exact number) (whole number)

The quick test

When facing a counting problem, ask: "If I rearrange the elements, do I get something different?"

What do you think?

You're forming a line at the store. Is this a permutation or combination?

What do you think?

You're drawing lottery numbers. Is this a permutation or combination?

Memory trick: "Permutation" and "Position" both start with P. If position matters, it's a permutation.

Classify 15 real-world scenarios as permutation or combination:

Permutation or Combination?

1 / 15

Choosing a 4-digit PIN code

Test your understanding

A password uses 4 distinct digits (0-9). How many passwords are possible? (whole number)

A team of 5 is chosen from 20 students. How many teams are possible? (whole number)

How many ways can you assign 3 different prizes to 3 of 8 contestants? (whole number)

True or False: C(n,k) is always less than or equal to P(n,k) (true or false)

What's next

Next: proving things about combinations without algebra.