Counting Principles & Probability (Grade 12)

The fundamental counting principle, arrangements with n!, handling items that must stay together or that repeat, and using counting to work out a probability.

1The fundamental counting principle

If one choice can be made in m ways and the next in n ways,
the two together can be made in m × n ways.

Worked ExampleHow many 4-digit PINs are there if digits may repeat? And if they may not?

1
With repeats: 10 × 10 × 10 × 10 = 10 000.
Every position has all ten digits available.
2
Without repeats: 10 × 9 × 8 × 7 = 5 040.
Each digit used removes one option from the next position.

2Arrangements: n!

n! = n × (n − 1) × … × 2 × 1

0! = 1

n different objects can be arranged in a row in n! ways. So 5 books arrange in 5! = 120 ways, and 7 people in 7! = 5 040 ways.

3When some items must stay together

💡

Tie them into a single block, arrange everything including the block, then arrange inside the block and multiply.

Worked Example4 boys and 3 girls sit in a row. In how many ways can they sit if the girls sit together?

1
Treat the 3 girls as one block. That leaves 4 boys + 1 block = 5 items.
2
Arrange the 5 items: 5! = 120.
3
Arrange the girls inside the block: 3! = 6.
The block is not fixed internally.
4
Total = 120 × 6 = 720.

4Counting to get a probability

P(event) = number of favourable arrangementstotal number of arrangements

Worked ExampleFor the 7 learners above, what is the probability that the girls sit together?

1
Favourable = 720 (from above).
2
Total arrangements of 7 people = 7! = 5 040.
No restriction.
3
P = 7205 040 = 17.
≈ 0,143.

5Repeated letters

arrangements of n letters with a letter repeated r times = n!r!

Worked ExampleHow many arrangements are there of the letters of the word LETTER?

1
There are 6 letters, but E appears twice and T appears twice.
2
6!2! × 2! = 7204.
Divide by the factorial of each repeat count.
3
= 180.

⚠️

Forgetting to divide by the repeats gives 720, which counts every identical-looking arrangement several times over.

6Tree diagrams still apply

A screening test: 20% of people have the disease; the test is 90% accurate on them and gives a false positive 20% of the time.

P(tests positive) = (15 × 910) + (45 × 210) = 950 + 850 = 1750 = 0,34.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
How many 4-digit PINs can be formed if digits may repeat?
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10⁴ = 10 000
2
How many 4-digit PINs can be formed if no digit may repeat?
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10 × 9 × 8 × 7 = 5 040
3
In how many ways can 5 different books be arranged on a shelf?
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5! = 120
4
4 boys and 3 girls sit in a row. In how many ways can they sit if the girls sit together?
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5! × 3! = 120 × 6 = 720
5
For those 7 learners, calculate the probability that the girls sit together.
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7205 040 = 17
6
How many arrangements are there of the letters of the word LETTER?
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6!2!×2! = 180
7
Write down the value of 0!
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1
8
Using the tree diagram below, calculate the probability that a person tests positive.
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15×910 + 45×210 = 0,34