£14 Million, who do you choose?
How many valid fantasy-league squads exist once budget and position limits are applied?
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
Combinatorics gives you a way to count possibilities without listing every one of them: how many passwords are possible, how many ways five people can sit around a table, how many 3-topping pizzas you can build from 10 toppings. A fantasy football squad is a counting problem with several constraints layered on top of each other — a fixed number of goalkeepers, defenders, midfielders and forwards, drawn from a much larger player pool, inside a fixed budget. This exploration works through the counting rules that make problems like this tractable (permutations, combinations, the multiplication principle) and then applies them to your own squad: first without constraints, then with position quotas, then with a budget cap.
Guiding Questions
- What is the difference between a permutation and a combination, and which one applies to picking a fantasy squad?
- A squad needs 2 goalkeepers, 5 defenders, 5 midfielders and 3 forwards from a pool of, say, 40 players. How many possible squads are there before you apply a budget?
- Each player has a price, and your total squad must fit under a fixed budget. How does that constraint cut down the number from your first answer?
- Pascal's triangle encodes combination counts directly. Where do the entries relevant to your squad-selection problem sit in the triangle?
- How does the multiplication principle let you count position by position, rather than listing every possible squad?