Systematic counting & combinatorics
Count possibilities systematically; use tables/tree diagrams to enumerate outcomes.
Some questions ask not “how much” but “how many ways” — how many outfits, how many routes, how many three-letter codes. Combinatorics is the art of counting those possibilities without missing any and without counting one twice.
What it means
Suppose you have 3 T-shirts and 2 pairs of shorts, and you want to know how many different outfits you can make. Guessing is risky: you might forget one. The safe method is to be systematic — to list possibilities in a fixed order so every combination appears exactly once.
Two tools make this reliable.
A tree diagram starts from one choice and branches out for the next. Pick a T-shirt (3 branches); from the end of each branch, pick shorts (2 more branches each). Count the tips of the tree: 3 × 2 = 6 outfits.
A table does the same job in a grid: T-shirts down the side, shorts across the top. Each cell is one outfit, so the number of outfits is rows × columns.
Both reveal the counting principle (also called the multiplication principle): if one choice can be made in a ways and, for each of those, a second choice can be made in b ways, then the two choices together can be made in a × b ways. It extends to more stages — you just keep multiplying.
The key idea is that being organised is what makes counting trustworthy. A tree or table is a promise that you have looked at every case once.
Worked examples
1. Ice-cream (flavour then cone). 3 flavours, 2 cones. 3 × 2 = 6. The tree:
Vanilla —— Wafer, Sugar
Choc —— Wafer, Sugar
Mint —— Wafer, Sugar2. Two coin flips. Each flip is Heads or Tails: 2 × 2 = 4 outcomes — HH, HT, TH, TT. Listing them in that order guarantees none are missed.
3. Three-digit lock, digits 1–3, no repeats. First digit: 3 choices. Second: 2 left. Third: 1 left. 3 × 2 × 1 = 6 codes: 123, 132, 213, 231, 312, 321.
The generative-art connection
An array of dots is combinatorics you can see. Line up 3 dots across and 2 down and you have drawn a 3 × 2 grid — every dot marks one outfit, one cell of the table. The picture is the multiplication.
The dot-multiplier tool makes this vivid: a single dot fans out into a symmetric burst of dots, and as you change the count the whole spray rearranges. Counting how many dots appear, and how they spread evenly, is the same “how many ways” question made visual. A tree diagram is itself a branching artwork — each new choice splits every path, so the drawing grows like a fractal, doubling or tripling at every level.
Common misconceptions
Adding instead of multiplying. With 3 shirts and 2 shorts, the answer is not 3 + 2 = 5. You add when choosing one thing from separate piles (shirt OR shorts); you multiply when combining choices (shirt AND shorts).
Forgetting that choices can shrink. In the no-repeats lock, the second digit had only 2 options, not 3 — the first pick used one up. Watch whether each stage keeps its full set of options or loses one.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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A café sells `4` kinds of pancake and `3` toppings. You pick one pancake and one topping. Draw a tree (or make a table) and say how many different pancake-with-topping orders you could make.
Answer
`4 × 3 = 12` different orders. A tree has 4 first branches, each splitting into 3, giving 12 tips; a table of 4 rows × 3 columns has 12 cells. You multiply because you choose a pancake AND a topping.
Art hook Canvas grid: 4 rows of dots × 3 columns, each dot a colour blend of its row-hue and column-hue. Tapping a cell lights the matching row and column so a child sees each order as one lit dot; a counter shows 12.
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A robot passes `3` junctions in a row, and at each junction it goes Left or Right. List all the paths in order, say how many there are, and explain how you made sure you missed none.
Answer
`2 × 2 × 2 = 8` paths. Listed systematically: LLL, LLR, LRL, LRR, RLL, RLR, RRL, RRR. Keeping the first letter fixed while the later letters run through every option guarantees each path appears exactly once and none is missed.
Art hook A binary tree grows down the canvas: one dot splits into 2, then 4, then 8, each branch angled left or right. Colour each of the 8 endpoint dots by its path (L adds a blue tint, R adds a red tint) so the endpoints fan out symmetrically.
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You have letter tiles A, B and C. Using each letter exactly once, how many different 3-letter arrangements can you make? List them in a sensible order.
Answer
`3 × 2 × 1 = 6` arrangements: ABC, ACB, BAC, BCA, CAB, CBA. There are 3 choices for the first letter, then 2 left, then 1 left. Fixing the first letter and listing the rest keeps the list complete and repeat-free.
Art hook Place 3 dots on a triangle; animate a line visiting them in each of the 6 orders, leaving a coloured trail per order. The 6 traced paths overlaid make a symmetric star pattern.
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A meal deal has `3` mains, `4` drinks and `2` desserts, and you choose one of each. Without drawing the whole tree, work out how many different meal deals are possible and explain your reasoning.
Answer
`3 × 4 × 2 = 24` meal deals. By the counting principle you multiply the number of choices at each stage: 3 ways for the main, times 4 for the drink, times 2 for the dessert.
Art hook Three concentric rings of dots (3 spokes, then 4, then 2). A line spirals outward picking one dot per ring; each complete spiral is one of the 24 meals, and running all 24 fills the canvas with a woven rosette.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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You have a red hat and a blue hat, and a red scarf and a blue scarf. You wear one hat and one scarf. List every hat-and-scarf combination you could wear.
Answer
`2 × 2 = 4` combinations: red hat + red scarf, red hat + blue scarf, blue hat + red scarf, blue hat + blue scarf.
Art hook Two dots (hats) each connected by lines to two dots (scarves): the 4 connecting lines are the 4 outfits. Colour each line by mixing its two endpoint colours.
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A spinner has `3` colours and a coin lands Heads or Tails. You spin once and flip once. Fill in a table with the 3 colours down the side and Heads/Tails across the top. How many cells does the table have?
Answer
`3 × 2 = 6` cells, so 6 outcomes. Each cell is one (colour, Heads-or-Tails) pair.
Art hook A 3×2 grid of cells on canvas; each cell shows its colour dot, striped if it is the 'Tails' column. Hovering a cell shows its pair on screen.
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Sam makes a sandwich with one bread (white or brown) and one filling (cheese, ham or jam). Draw a tree diagram and count the sandwiches.
Answer
`2 × 3 = 6` sandwiches: white-cheese, white-ham, white-jam, brown-cheese, brown-ham, brown-jam. The tree has 2 first branches, each splitting into 3.
Art hook A tree with 2 trunk-branches (bread colours) each fanning into 3 twigs (filling colours); the 6 leaf-dots sit in a neat row, each a two-tone blend.
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Spot the mistake: Mia says, 'I have `5` shirts and `2` hats, so I can make `5 + 2 = 7` different shirt-and-hat looks.' Is she right? If not, fix it.
Answer
She is wrong. She should multiply, not add, because she picks a shirt AND a hat: `5 × 2 = 10` looks. You add only when choosing a single item from separate piles (a shirt OR a hat).
Art hook Left panel shows 5+2=7 as two separate short rows of dots; right panel shows a 5×2 grid of 10 dots. A toggle flips between 'OR' (add) and 'AND' (multiply) to show why the grid is the right picture.
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How many two-digit numbers can you build using the digits `1`, `2` and `3` if a digit may be used twice (so 22 is allowed)? List them.
Answer
`3 × 3 = 9`: 11, 12, 13, 21, 22, 23, 31, 32, 33. The first digit has 3 choices and the second digit still has 3 choices because repeats are allowed.
Art hook A 3×3 grid where cell (row r, column c) shows the number formed by first digit r and second digit c — so the 9 numbers 11..33 fill a perfect square lattice, one per dot.
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Now build two-digit numbers from `1`, `2`, `3` but with no digit repeated. How many are there, and how is this different from the last question?
Answer
`3 × 2 = 6`: 12, 13, 21, 23, 31, 32. The second digit now has only 2 choices because the first pick used one digit up — so there are fewer than the 9 before.
Art hook Draw the same 3×3 number lattice, then grey out the three diagonal dots (11, 22, 33); the 6 remaining dots are the 'no-repeat' pairs, and the greyed diagonal shows exactly which repeats were removed.
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A pizza comes in Small, Medium or Large, with a Thin or Thick crust, and Red or White sauce. You choose one of each. How many different pizzas can be ordered? Explain each factor.
Answer
`3 × 2 × 2 = 12` pizzas. 3 sizes, times 2 crusts, times 2 sauces — multiply the choices at each of the three stages.
Art hook Dot size = pizza size, dot ring thickness = crust, dot fill hue = sauce. Lay out all 12 dots in a grid so every size/crust/sauce combination appears exactly once.
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Reason it out: at a class raffle each ticket has one letter (A or B) then one digit (`1`, `2`, `3` or `4`), like A3. Are there enough different tickets for `10` children to each get a unique one?
Answer
`2 × 4 = 8` different tickets (A1–A4, B1–B4). That is only 8, which is fewer than 10, so no — there are not enough unique tickets for 10 children.
Art hook A 2×4 grid of ticket-dots labelled A1..B4. A counter compares 8 against the target 10; if children outnumber tickets, the extra spots flash to show the shortfall.
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A café offers `4` sandwiches and `5` drinks for lunch, and you choose one sandwich and one drink. Without listing them all, how many lunches are possible? Then say how many rows and columns its table would have.
Answer
`4 × 5 = 20` lunches. The table would have 4 rows (sandwiches) and 5 columns (drinks), giving 4 × 5 = 20 cells.
Art hook A 4×5 grid of dots; sweeping a highlight down one row shows 'this sandwich with each drink', making the ×5 visible five dots at a time until all 20 are lit.
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Open-ended: invent your own 'how many ways' question that has exactly `12` possibilities, using two or three stages of choices. Show the multiplication that gives 12.
Answer
Many answers work, e.g. 3 hats × 4 scarves = 12, or 2 shoes × 3 shirts × 2 hats = 12, or 6 ice-creams × 2 cones = 12. Correct if the stage-choices multiply to 12 and the context is one item chosen per stage.
Art hook A builder tool: the child sets 2 or 3 sliders (choices per stage); the canvas auto-draws the matching grid or tree of dots and shows the product live, celebrating when it equals 12.
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A tree diagram for choosing a fruit then a colour of bag has `15` tips at the bottom. If there were `5` fruits, how many bag colours were there? Explain how you undid the multiplication.
Answer
3 bag colours, because `5 × ? = 15` means `15 ÷ 5 = 3`. Division undoes the counting-principle multiplication to find the missing number of choices.
Art hook Draw the tree backwards: start from 15 leaf-dots grouped into 5 clusters; the tool reveals each cluster has 3 dots, so the branching factor 3 lights up.