Problem of the Week
Problem A and Solution
Summer Selections

Problem

Robbie is trying to simplify his summer wardrobe. He always wears a pair of shorts, a shirt, and a hat. He has:

• $3$ pairs of shorts in the colours of red, grey, and yellow

• $3$ shirts in the colours of red, white, and green

• $3$ hats in the colours of yellow, purple, and white

How many different combinations of a pair of shorts, a shirt, and a hat can he put together so that each item is of a different colour?

Solution

One way to solve this problem is to create a tree that shows all possible combinations of shorts, shirts, and hats that Robbie can wear. Then we can count the number of leaves of the tree (items in the tree that do not have any branches leading away from them) to determine the number of possible colour combinations.

When creating the tree, it is important that we avoid duplicating colours in each outfit. We can check this by following a path from the root of the tree (“Outfits”) to each leaf and ensuring there are no duplicate colours. Each path from the root to a leaf describes a colour combination of a single outfit.

For example red shorts $\to$ green shirt $\to$ purple hat is one outfit and grey shorts $\to$ red shirt $\to$ white hat is another outfit. There are no duplicate colours in those outfits.

Here is tree showing those combinations:

Since there are $18$ leaves in this diagram, there are $18$ possible outfits Robbie could wear without duplicating colours.