Steply

Counting choices

Probability · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Count the choices
  2. Draw a tree
  3. Multiply
  4. Check

The steps for 2 shirts and 3 pants, how many outfits. Every solution names its steps, so you always know what comes next.

Worked examples

2 shirts and 3 pants, how many outfits Easy

The question

Question: There are 2 shirts and 3 pants.

How many outfits can be made with one of each?

1 Count the choices

2 Draw a tree

3 Multiply

Each of the 2 shirts goes with 3 pants.

So we multiply: 2 × 3.

outfits = 2 × 3

This is the counting principle: multiply the number of choices.

counting principle:
multiply the choices

2 groups of 3: that is 2 times 3.

3 + 3 = 2 × 3

Careful!

We do not add the choices.

Every choice goes with every other one, so we multiply.

Careful: not 2 + 3 = 5 ✗

2 times 3 is 6.

2 × 3 = 6

So there are 6 outfits.

outfits = 6

4 Check

Let's check: we count the ends of the tree.

There are 6 ends, one for each.

So the answer is right!

6 ends = 6 outfits

Let's look back.

First: count the choices.

Then: draw a tree.

Then: multiply.

Last: check.

Answer
outfits = 6

▶ Watch it solved

3 shirts and 4 pants, how many outfits Medium

The question

Question: There are 3 shirts and 4 pants.

How many outfits can be made with one of each?

1 Count the choices

2 Multiply

Each of the 3 shirts goes with 4 pants.

So we multiply: 3 × 4.

outfits = 3 × 4

This is the counting principle: multiply the number of choices.

counting principle:
multiply the choices

3 groups of 4: that is 3 times 4.

4 + 4 + 4 = 3 × 4

Careful!

We do not add the choices.

Every choice goes with every other one, so we multiply.

Careful: not 3 + 4 = 7 ✗

3 times 4 is 12.

3 × 4 = 12

So there are 12 outfits.

outfits = 12

3 Check

Let's check: 12 divided by 3 is 4.

That gives back the 4 choices.

So the answer is right!

12 ÷ 3 = 4

Let's look back.

First: count the choices.

Then: multiply.

Last: check.

Answer
outfits = 12

▶ Watch it solved

A code has 3 digits, how many codes Harder

The question

Question: A code has 3 digits (0 to 9).

How many codes are there?

1 Count the choices

2 Multiply

Every choice goes with every other choice.

So we multiply them all: 10 × 10 × 10.

codes = 10 × 10 × 10

This is the counting principle: multiply the number of choices.

counting principle:
multiply the choices

Careful!

We do not add the choices.

Every choice goes with every other one, so we multiply.

Careful: not 10 + 10 + 10 = 30 ✗

10 times 10 is 100.

100 times 10 is 1000.

10 × 10 = 100
100 × 10 = 1000

So there are 1000 codes.

codes = 1000

3 Check

Let's check in another order.

First 10 times 10 is 100.

10 × 10 = 100

Then 10 times 100 is 1000.

The same.

So the answer is right!

10 × 100 = 1000

Let's look back.

First: count the choices.

Then: multiply.

Last: check.

Answer
codes = 1000

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. 2 shirts and 3 pants, how many outfits
    Answer
    outfits = 6
  2. 3 shirts and 2 pants, how many outfits
    Answer
    outfits = 6
  3. there are 2 drinks and 4 snacks, how many meals
    Answer
    meals = 8
  4. flip a coin 2 times, how many outcomes
    Answer
    outcomes = 4
  5. flip a coin 3 times, how many outcomes
    Answer
    outcomes = 8
  6. 2 hats and 4 scarves, how many outfits
    Answer
    outfits = 8
  7. a menu has 2 starters and 4 mains, how many meals
    Answer
    meals = 8
  8. 4 shirts and 2 pants, how many outfits
    Answer
    outfits = 8

Medium

  1. 3 shirts and 4 pants, how many outfits
    Answer
    outfits = 12
  2. 5 shirts and 3 pants, how many outfits
    Answer
    outfits = 15
  3. a menu has 3 starters, 5 mains and 2 desserts, how many meals
    Answer
    meals = 30
  4. 2 breads, 3 meats and 4 sauces, how many sandwiches
    Answer
    sandwiches = 24
  5. roll a die and flip a coin, how many outcomes
    Answer
    outcomes = 12
  6. roll two dice, how many outcomes
    Answer
    outcomes = 36
  7. flip a coin 5 times, how many outcomes
    Answer
    outcomes = 32
  8. 6 colours and 4 sizes, how many shirts
    Answer
    shirts = 24

Harder

  1. a code has 3 digits, how many codes
    Answer
    codes = 1000
  2. a code has 4 digits, how many codes
    Answer
    codes = 10000
  3. a code has 3 different digits, how many codes
    Answer
    codes = 720
  4. a pin has 4 different digits, how many pins
    Answer
    PINs = 5040
  5. a plate has 2 letters and 3 digits, how many plates
    Answer
    plates = 676000
  6. a plate has 3 letters and 2 digits, how many plates
    Answer
    plates = 1757600