The four counting ideas
Every question in this unit is one of four things. Once you can spot which, the formula is the easy part.
The fundamental counting principle
The idea: if one decision can be made in m ways and a second in n ways, the two together can be made in m × n ways. Draw a box for each decision, write the number of choices in it, and multiply.
This is the foundation everything else sits on. The trick is not the multiplying — it is noticing how many separate decisions there actually are.
Worked example
A restaurant special includes one appetizer, one main and one dessert. There are 2 appetizers, 5 mains and 3 desserts. How many different meals are possible?
Three decisions, so three boxes:
When choices run out
If something cannot be reused, the number in each box drops by one as you go. A 4-digit PIN using only the digits 1–5 with no repeats is 5 × 4 × 3 × 2 = 120, not 5⁴.
Watch out: adding instead of multiplying. With 3 shirts and 4 pairs of pants, 3 + 4 = 7 counts the pieces of clothing. The outfits are 3 × 4 = 12.
Permutations
The idea: order matters. ₙPᵣ = n! ÷ (n − r)!. To arrange n objects where some are identical, divide n! by the factorial of each repeat count.
A permutation is an arrangement. Swapping two things gives you a different permutation — which is exactly what makes it different from a combination.
Factorials first
n! means multiply n by every whole number below it, down to 1. So 5! = 5 × 4 × 3 × 2 × 1 = 120. It is not 5 × 5, and it does not stop partway.
Worked example
Evaluate ₈P₂.
The formula gives 8! ÷ 6!. Written out, everything from 6 downwards appears on the top and the bottom, so it cancels:
When letters repeat
BOOK has 4 letters, so 4! = 24 — except the two O's are identical, so every word got counted 2! times. Divide it out: 24 ÷ 2 = 12. MISSISSIPPI has I four times, S four times and P twice, so it is 11! ÷ (4! 4! 2!) = 34 650.
Watch out: treating repeated letters as if they were different. If swapping two of them gives the same word, you have counted it twice.
Combinations
The idea: order does not matter. ₙCᵣ = n! ÷ [(n − r)! r!]. Handy link: ₙCᵣ = ₙPᵣ ÷ r!, because each group can be ordered r! ways.
A combination is a group. Picking Ana then Ben gives the same group as picking Ben then Ana, so it should only be counted once.
Worked example
Evaluate ₅C₂ — choosing 2 things from 5.
₅C₂ = 5! ÷ (3! × 2!) = 120 ÷ 12 = 10. You can see all ten at once by drawing 5 points and joining every pair:
Choosing from two groups
If a team of 4 needs exactly 2 girls from 7 and 2 boys from 5, choose each group separately and multiply: ₇C₂ × ₅C₂ = 21 × 10 = 210.
Watch out: using a permutation when order does not matter. ₅P₂ = 20 but ₅C₂ = 10 — a combination is always the smaller of the two.
The binomial theorem
The idea: (a + b)ⁿ has n + 1 terms. The coefficients are row n of Pascal's triangle, which are also ₙC₀, ₙC₁, …, ₙCₙ. The general term is ₙCₖ aⁿ⁻ᵏ bᵏ.
Expanding a bracket to a power by hand gets miserable fast. Pascal's triangle hands you the coefficients for free.
Finding one particular term
You rarely need the whole expansion — usually just one term. Use ₙCₖ aⁿ⁻ᵏ bᵏ and solve for k.
For the coefficient of x³ in (x + 2)⁵: you need x³, so n − k = 3, meaning k = 2. The term is ₅C₂ x³ 2² = 10 × 4 = 40x³.
For the constant term in (x + 1/x)⁶: the powers give x⁶⁻ᵏ × x⁻ᵏ = x⁶⁻²ᵏ. A constant needs the power to be zero, so k = 3, and the term is ₆C₃ = 20.
Watch out: forgetting to raise the whole term to the power. In (2x + 3)³ the first term is (2x)³ = 8x³, not 2x³. Bracket it before you cube it.
Which one do I need?
Ask one question: if I swap two of these around, is it different?
- Yes, it's different → order matters → permutation. Look for roles, rankings, positions, passwords, or a row.
- No, it's the same → order does not matter → combination. Look for teams, committees, groups, hands of cards, or toppings.
- Several separate decisions in a row → multiply the boxes → counting principle.
- A bracket raised to a power → binomial theorem.
Gold, silver and bronze is a permutation. Three students going on a field trip is a combination. Same numbers, different answer.