Counting
Multiply choices, arrange things in order, handle repeats.
Pen and paper is fine · no calculator needed why?
Opens at level 15. You're level 1. You can read and practice here now.
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.
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.
worked example
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.
worked example
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.
worked example
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
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?
Answer: 49
- Each choice multiplies the options: 7 × 7 = 49.
Factorial ratios
worked example
What is 6! ÷ 3!?
Answer: 120
- Everything from 3 down cancels, leaving 6 × 5 × 4 = 120.
Ordered picks
worked example
6 runners race. In how many ways can first place and second place be awarded, with no ties?
Answer: 30
- 6 choices for first place and 5 choices for second place: order matters, so multiply.
- 6 × 5 = 30.
Arrangements with repeats
worked example
How many different arrangements are there of the letters in LITTLE?
Answer: 180
- 6! = 720 orders, but rearranging the two Ls or the two Ts among themselves gives the same word.
- 720 ÷ (2! × 2!) = 720 ÷ 4 = 180.