Counting outcomes.

Step 1 of 4

A tree of choices

In the morning you pick a shirt out of $3$⁠, then trousers out of $2$⁠. How many different outfits are there?

The board draws a tree diagram: from the dot at the top a branch leaves for each shirt, and from each shirt a branch leaves for each pair of trousers. Every path from the top to the bottom row is one outfit, and the bottom row has $3 \times 2 = 6$⁠ ends.

This is the multiplication principle: when you choose in stages, and every option at one stage meets every option at the next, the number of outcomes is the product of the numbers of options.

Now add a hat. Predict how many outfits there will be, then watch the tree grow.

Step 2 of 4

Two dice

When you roll two dice, the tree would split into six and then into six again. A table is handier here: the first die across, the second down, and in each square the pair that came up.

Notice that the square $(3,4)$⁠ and the square $(4,3)$⁠ are two different outcomes: $3$⁠ on the first and $4$⁠ on the second, or the other way round. Count the squares or use the multiplication principle.

Step 3 of 4

Coin tosses

A coin lands on heads (H) or tails (T). At every toss each path of the tree splits in two, so every extra toss doubles the number of paths.

Change Tosses and watch the bottom row. The number under each end is how many heads came up on that path.

Step 4 of 4

How many orders?

Three books, A, B and C, stand on a shelf. How many different orders can they stand in?

The first place has $3$⁠ books to choose from. Once you have chosen, $2$⁠ are left for the second place, and one book is left for the last. The tree splits into three, then into two, then a single branch: $3 \times 2 \times 1 = 6$$3 \times 2 \times 1 =$$6$⁠.

A product like this, a number times every whole number below it down to $1$⁠, is called a factorial and is written with an exclamation mark: $3! = 3 \times 2 \times 1 = 6$$3! =$$3 \times 2 \times 1 =$$6$⁠.