Counting
Multiply choices, arrange things in order, handle repeats.
Pen and paper is fine · no calculator needed why?
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the idea
Counting problems ask "how many ways", and most can be done without listing. Multiply the choices: 4 shirts and 3 pairs of trousers make 4 × 3 = 12 outfits, because every shirt goes with every pair.
more of the idea
Putting n different things in a row gives n × (n − 1) × … × 1 orders. That product is n factorial, written n followed by an exclamation mark. Picking only a few of them in order stops the product early. When some items are identical, swapping them makes no new arrangement, so you divide those swaps out.
techniques
Multiply the choices
- Count the options in each group.
- Multiply the counts. Every option in one group pairs with every option in the others.
A café offers 3 starters, 5 main courses and 2 desserts. How many different meals with one of each can you order?
- 3 × 5 × 2 = 30.
Answer: 30
Fill the slots in order
- Count the choices for the first slot, then the next, one fewer each time.
- Multiply as many numbers as there are slots.
- Factorial ratios work the same way: 9 factorial ÷ 6 factorial cancels everything from 6 down, leaving 9 × 8 × 7.
A club has 7 members. In how many ways can it choose a president, a vice-president and a secretary, if no one holds two posts?
- 7 choices for president, then 6, then 5.
- 7 × 6 × 5 = 210.
Answer: 210
Divide out the repeats
- Count all the orders as if every letter were different.
- For each repeated letter, divide by the factorial of how many times it appears.
How many different arrangements are there of the letters in GEESE?
- 5 letters: 5 × 4 × 3 × 2 × 1 = 120 orders.
- The three Es swap in 3 × 2 × 1 = 6 ways.
- 120 ÷ 6 = 20.
Answer: 20
watch out for
- Adding the options instead of multiplying. Every shirt goes with every pair of trousers.
- Treating 10 factorial ÷ 8 factorial as 2 factorial. Cancel everything from 8 down; what is left is 10 × 9.
- Dividing by the number of times a letter repeats instead of its factorial. Three Es means dividing by 6, not 3.
practice
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Multiply the choices
worked example
A café offers 7 starters and 7 main courses. How many different meals with one of each can you order?
- Each choice multiplies the options: 7 × 7 = 49.
Answer: 49
Factorial ratios
worked example
What is 6! ÷ 3!?
- Everything from 3 down cancels, leaving 6 × 5 × 4 = 120.
Answer: 120
Ordered picks
worked example
6 runners race. In how many ways can first place and second place be awarded, with no ties?
- 6 choices for first place and 5 choices for second place: order matters, so multiply.
- 6 × 5 = 30.
Answer: 30
Arrangements with repeats
worked example
How many different arrangements are there of the letters in LITTLE?
- 6! = 720 orders, but rearranging the two Ls or the two Ts among themselves gives the same word.
- 720 ÷ (2! × 2!) = 720 ÷ 4 = 180.
Answer: 180