Practice Maths

Combinatorics — Topic Review

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This review covers all five lessons in the Combinatorics topic: Multiplication and Addition Principles, Permutations, Combinations, Pascal’s Triangle and the Binomial Theorem, and Inclusion–Exclusion and Pigeonhole. Questions are exam-style and increase in difficulty.

Mixed Review Questions

  1. Fluency

    Q1 — Meal Combinations (Multiplication Principle)

    A set menu offers 3 entrees, 5 mains, and 4 desserts. How many different 3-course meals can be ordered?

  2. Fluency

    Q2 — Factorial Simplification

    Simplify without a calculator:   (a) 9!/7!    (b) 12!/(3! × 9!)

  3. Fluency

    Q3 — Arranging Books on a Shelf (Permutations)

    In how many ways can 5 different books be arranged on a shelf?

  4. Fluency

    Q4 — Choosing a Team (Combinations)

    In how many ways can a group of 3 be chosen from 8 people to serve on a committee (no assigned roles)?

  5. Fluency

    Q5 — Binomial Expansion (Pascal’s Triangle)

    Expand (x + 3)3 using the binomial theorem.

  6. Fluency

    Q6 — Survey Data (Inclusion–Exclusion)

    Of 60 people surveyed, 35 drink coffee, 28 drink tea, and 14 drink both. How many drink at least one of the two beverages?

  7. Understanding

    Q7 — Seating with a Restriction (Permutations)

    Seven people sit in a row. In how many ways can this be done if two particular people, Alex and Sam, must NOT sit next to each other?

  8. Understanding

    Q8 — Selecting a Mixed Group (Combinations)

    A committee of 6 is chosen from 8 men and 5 women. How many committees contain exactly 4 men and 2 women?

  9. Understanding

    Q9 — Specific Term in a Binomial Expansion

    Find the coefficient of x2 in the expansion of (3x − 2)5.

  10. Understanding

    Q10 — Three Subjects (Inclusion–Exclusion)

    In a group of 200 students: 110 study Biology, 85 study Chemistry, 70 study Physics. Pairwise: 40 study Bio and Chem, 30 study Bio and Phys, 25 study Chem and Phys. 15 study all three. How many students study none of the three subjects?

  11. Understanding

    Q11 — Arrangements with Repeated Letters (Permutations)

    How many distinct arrangements are there of the letters in STATISTICS?

  12. Problem Solving

    Q12 — At Least Restriction (Combinations)

    A hand of 5 cards is dealt from a standard 52-card deck. How many hands contain at least one ace? (There are 4 aces in the deck.)

  13. Problem Solving

    Q13 — Constant Term in a Binomial Expansion

    Find the constant term in the expansion of (2x2 + 1/x)9.

  14. Problem Solving

    Q14 — Pigeonhole Application

    A sequence of 101 integers is chosen from {1, 2, 3, …, 200}. Prove that at least two of the integers in the sequence sum to 201.

  15. Problem Solving

    Q15 — Combined Counting Problem

    A password is 6 characters long. It must contain at least one digit (0–9) and at least one letter (A–Z, 26 letters). Characters may repeat, and the order matters. How many valid passwords are there? (Hint: use complementary counting.)

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