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Combinatorics (Primary)

In brief

What is combinatorics? Combinatorics is the branch of mathematics that deals with counting — not just by listing, but by using smart rules and patterns to calculate the number of ways something can happen. We’ll explore questions like: • How many different outcomes are possible when making choices? • What’s the difference between arrangements (where order matters) and selections (where it doesn’t)? • How can we use multiplication, permutations, and combinations to solve these problems?

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Problem set (reference)

Class problems as we used them on this topic. For interactive practice with solutions and progress tracking, use Problems.cc when available.

  1. A toy shop sells 5 types of teddy bears and 3 types of toy cars. How many different teddy-and-car combinations can you choose?

  2. A toy shop sells 5 types of teddy bears, 3 types of toy cars, and 4 types of spinning tops. How many different choices are there if you pick:
    a) one teddy bear, one toy car, and one spinning top?
    b) any two different toys?

  3. How many four-digit numbers are there such that:
    a) only even digits are used?
    b) there is at least one even digit?
    (Hint: Even digits are 0, 2, 4, 6, 8.)

  4. You flip a coin three times. How many different possible outcomes (sequences of heads and tails) are there?

  5. Each square in a 2×2 grid is coloured either black or white. How many different ways are there to colour the grid?

  6. In the fictional Mumbo-Yumbo language, the alphabet has just two letters: A and U. A “word” can be any sequence of up to 5 letters. How many different words are there in the Mumbo-Yumbo dictionary?

  7. A football team has 11 players. In how many ways can you choose:
    a) a captain and a vice-captain (two different people)?
    b) two strikers (order doesn’t matter)?

  8. How many ways can you place:
    a) a black rook and a white rook on a chessboard?
    b) a black king and a white king, such that they don’t attack each other?
    (Reminder: Rooks attack any square in the same row or column. Kings attack all adjacent squares.)

  9. Charlie has 4 different-coloured counters. In how many different ways can he arrange them in a row?

  10. How many five-digit numbers can be formed using only odd digits, such that each digit is different?
    (Hint: Odd digits are 1, 3, 5, 7, 9.)

  11. Which are more numerous: seven-digit numbers that include the digit 1, or those that don’t contain the digit 1 at all?

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