Theory
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?
Problems
- 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?
- 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? - 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.) - You flip a coin three times. How many different possible outcomes (sequences of heads and tails) are there?
- Each square in a 2×2 grid is coloured either black or white. How many different ways are there to colour the grid?
- 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?
- 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)? - 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.) - Charlie has 4 different-coloured counters. In how many different ways can he arrange them in a row?
- 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.) - Which are more numerous: seven-digit numbers that include the digit 1, or those that don’t contain the digit 1 at all?