Numerical aptitude · Counting and chance
Permutations and combinations, made easy
💡 The idea
Fill the places one at a time and multiply the choices. If the order does not matter (a team, a handshake), divide by the number of orders.
1st: 5 choices→2nd: 4→3rd: 3→5 × 4 × 3 = 60
🧭 The method: 3 steps
- Draw the places (chairs, digits, letters) and write how many choices each has.
- Multiply them.
- Order does not matter? Divide by the orders of the chosen ones (2! for pairs, 3! for three…). Repeated letters? Divide by their orders too.
🎬 Every kind of question, in pictures
Pick a kind. Guess the answer first, then watch the picture work it out. Tap any step to see it in the picture.
Loading the pictures…
✏️ Worked examples
Example 1. 4 friends in a row?
- 4 × 3 × 2 × 1 = 24
⚡ Shortcut: 4! = 24.
Example 2. Team of 2 from 5?
- In order: 5 × 4 = 20
- A–B and B–A are the same team: ÷ 2
- 10 teams
⚡ Shortcut: 5C2 = 10.
Example 3. Arrange the letters of LEVEL
- 5 letters: 5! = 120
- Two Ls and two Es: ÷ (2 × 2)
- = 30
⚠️ Where people go wrong
- Adding instead of multiplying the choices.
- For teams and handshakes, forgetting to divide by the orders.
- Round table: fix one person first, (n − 1)!
📌 Remember
- Row: n!
- Choose r of n: n!/(r!(n−r)!)
- Round table: (n−1)!
- Handshakes: n(n−1)/2