Arrangements: Factorials as Trees

Lesson 2 of 11 · 7 min · Free, no account needed

What you will learn

An arrangement (permutation) is an order. n different things can be ordered in n! ways.
n! = n × (n - 1) × ... × 2 × 1: each place has one fewer choice than the place before.
Factorials grow fast: 10! is 3,628,800 and 14! is more than the number of people on Earth.
Filling only k places stops the product early: 3 medals from 8 runners is 8 × 7 × 6 = 336.

10 friends line up for a photo.
How many different orders are there? More or fewer than people on Earth? Drag to your bet.
Game: photo booth. Snap every different order of three friends.
Tap two friends to swap them, then Snap. A repeat photo costs a star.
Four friends. Grow the tree one spot at a time.
For each spot, tap how many friends could stand there.
Find the shortest line with more orders than people on Earth.
Drag the lever to add friends. Watch the bar.
Each new friend multiplies the count by the new line length, so the bar jumps further every time: 13! is 6.2 billion, 14! is 87 billion.
Orders are called arrangements, or permutations.
An arrangement (or permutation) is one order of things.
n different things have n! = n × (n - 1) × ... × 1 arrangements. Say "n factorial".
5! = 5 × 4 × 3 × 2 × 1 = 120.
6 runners race. How many different finishing orders are there?
First place: 6 options. Second: 5 left, then 4, 3, 2, 1. 6 × 5 × 4 × 3 × 2 × 1 = 720.
8 runners, but only gold, silver and bronze.
Set how many runners could still win each medal.
Only three places are filled, so the product stops after three factors: 8 × 7 × 6 = 336.
A language model reads words in order.
Rearrange the words into your own order and watch the counts.
n things have n! orders.
Shuffle the ten friends. Each shuffle is one of 10! line-ups.

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