Math Detours · Grade 5 · Around the corner · 13

Three shirts, four trousers

Twelve outfits, and a tree diagram shows every one. The same number of choices on every branch means you can multiply.

Name ____________________Date ____________

Tonight's idea

You have 3 shirts: red, white and yellow. You have 4 pairs of trousers: jeans, shorts, cords and tracksuit bottoms. An outfit is one shirt with one pair of trousers. How many outfits can you make?

start red J S C T white J S C T yellow J S C T 3 shirts, and 4 ends under each one: 12 ends
J jeans, S shorts, C cords, T tracksuit bottoms. The orange path is the white shirt with shorts.

The picture is a tree diagram. It splits each time you choose. First it splits 3 ways, one branch for each shirt. Then every shirt splits 4 ways, one branch for each pair of trousers.

Follow any path from the top to the bottom and you have picked an outfit. So the number of outfits is the number of ends. There are 3 groups of 4 ends, and \(3 \times 4 = 12\).

Why multiply and not add? Every shirt can go with every pair of trousers. Each shirt brings 4 outfits, so every shirt branch has 4 ends.

That same number on every branch matters. If the red shirt could go with only jeans and shorts, but the other shirts could still go with all 4 pairs, the tree would have \(2 + 4 + 4 = 10\) ends.

Same number of next choices on every branch: multiply. Different numbers: count the ends, or add the branch totals.

Watch one: the sandwich shop

You pick 1 bread from 2, then 1 filling from 3, then 1 drink from 2. How many different lunches are there?

The tree splits 2 ways for the bread. Each branch splits 3 ways for the filling. That makes \(2 \times 3 = 6\) sandwiches. Each of those 6 ends then splits 2 ways for the drink, so the branches stay even.

\[2 \times 3 \times 2 = 12 \text{ lunches}\]

Your turn

  1. An ice cream shop has 5 flavours and 3 toppings. You choose one flavour and one topping. How many different ice creams can you order?
  2. You have 2 hats (a cap and a beanie) and 3 scarves (red, green and blue). Draw the tree diagram. Then list every hat and scarf pair.
  3. A bike lock has 3 wheels. Each wheel shows one digit from 0 to 9. (a) How many different digits can one wheel show? (b) How many different codes can the lock show?
  4. An ice cream cone has two scoops, one on top of the other. There are 4 flavours. Chocolate on top of vanilla is a different cone from vanilla on top of chocolate. (a) How many cones are there if both scoops may be the same flavour? (b) How many if the two scoops must be different? (c) What changed in the tree?

Stretch: line up for a photo

Three friends stand in a row for a photo. How many different orders are there? Think of the choices for the left place, then the middle, then the right. Now try four friends.

Answers

  1. \(5 \times 3 = 15\) ice creams.
  2. The tree splits 2 ways, then each branch splits 3 ways. \(2 \times 3 = 6\) pairs: cap and red, cap and green, cap and blue, beanie and red, beanie and green, beanie and blue.
  3. (a) 10 digits. Do not forget 0. (b) \(10 \times 10 \times 10 = 1000\) codes, from 000 to 999.
  4. (a) 4 choices for the bottom scoop, then 4 for the top: \(4 \times 4 = 16\). (b) 4 choices for the bottom scoop, then only 3 flavours are left for the top: \(4 \times 3 = 12\). (c) The second split has 3 branches, not 4. The first choice used up one flavour.

Stretch. Three friends: 3 choices for the left place, then 2 friends are left for the middle, then 1 for the right. \(3 \times 2 \times 1 = 6\) orders. Four friends: \(4 \times 3 \times 2 \times 1 = 24\) orders. Next year's sheet takes this further: grade 6, factorials.

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