Lesson 42: Finding Permutations and Combinations (Exploration: Pascal's Triangle and Combinations)

New Concept

A permutation is a selection of items where order is important.

Why it matters

Mastering permutations and combinations is your first step into combinatorics, the art of sophisticated counting. This skill is crucial for probability, computer science algorithms, and making strategic decisions in complex scenarios.

What’s next

Next, you’ll learn the formal notation and formulas to precisely calculate the number of possible arrangements and selections.

The factorial of a positive integer $n$, denoted as $n!$, is the product of all positive integers up to and including $n$. By definition, zero factorial is 1.

Evaluate $4!$: $4! = 4 \times 3 \times 2 \times 1 = 24$.
Evaluate $\frac{8!}{6!}$: $\frac{8 \times 7 \times 6!}{6!} = 8 \times 7 = 56$.
Evaluate $\frac{7!}{3!(7-3)!}$: $\frac{7!}{3!4!} = \frac{7 \times 6 \times 5 \times 4!}{3 \times 2 \times 1 \times 4!} = \frac{210}{6} = 35$.

Think of a factorial as a countdown multiplication party! To find $5!$, you just multiply 5 by every whole number below it until you hit 1. It’s a super quick way to find the total possible arrangements of a full set of distinct items. Don't forget the weird guest at the party: $0!$ is always 1!

A permutation is an arrangement of objects where the order is important. The number of permutations of $n$ objects taken $r$ at a time is:

How many ways can a President and VP be chosen from 8 candidates? $P(8, 2) = \frac{8!}{(8-2)!} = \frac{8!}{6!} = 56$.
Find the distinguishable permutations of the letters in 'CHEESE': $\frac{6!}{3!} = \frac{720}{6} = 120$.
How many 3-digit lock codes can be made from digits 1-7 if no digit repeats? $P(7, 3) = \frac{7!}{(7-3)!} = 210$.

Permutations are all about position! Think of it like a race: getting 1st, 2nd, and 3rd place is a different outcome than the same three people in a different order. Use this when arranging things where the sequence is critical, like letters in a password or runners finishing a race. Order is everything here!

A combination is a selection of items where order does not matter. The number of combinations of $n$ objects taken $r$ at a time is:

How many 3-topping pizzas can be made from 8 available toppings? $C(8, 3) = \frac{8!}{3!(8-3)!} = 56$.
How many ways can you choose a 4-person team from 10 students? $C(10, 4) = \frac{10!}{4!(10-4)!} = 210$.
How many 5-card hands can be drawn from a 52-card deck? $C(52, 5) = \frac{52!}{5!47!} = 2,598,960$.

Combinations are for when you just want a group, and the order you pick them in is irrelevant. Think of making a fruit salad: putting in an apple then a banana is the same salad as a banana then an apple. Use this when you're selecting a committee, a hand of cards, or pizza toppings!