Slides for class
To project in class and work in groups.
Combinatorics from scratch · I. Counting without counting
Slides (22 slides): the product rule with tables and trees, counting digit by digit, permutations and factorials, and words with repeated letters. Challenges with solutions at the end.
Combinatorics from scratch · II. Binomial coefficients
Slides (26 slides): Ana’s routes (a Gardner problem), the sum rule, Pascal’s triangle, bijections and binomial coefficients, with their formula and two proofs of Pascal’s rule.
Combinatorics from scratch · III. Place your bets
Slides (27 slides): classic bets to vote on before calculating. The Chevalier de Méré, Galileo’s three-dice problem, the birthday problem, the lottery against 24 heads, Newton and Pepys, and Pascal and Fermat’s problem of points.
Teaching objectives
A combinatorics lab for lower secondary (ages 14-16). Everything revolves around one question: Ana wants to walk to school by a different route every day — how many days can she do it? Ana’s town is a grid of streets where students draw routes by tapping the crossings, and the animations show how routes add up crossing by crossing until the grid turns round and becomes Pascal’s triangle.
Rooms 1-4 review powers and factorials; 5-9 are Ana’s problem; 10-13 lead to choosing without order; 14-16 introduce the triangle and binomial coefficients with animated combinatorial proofs; and 17-20 go further: subsets, patterns in the triangle, Sierpinski and Newton’s binomial theorem.
What students learn
- The product rule: n^k (coins, dice, PIN codes) and n! (people in a row), with the idea that once the first is fixed there are (n − 1)! ways left.
- Items that cannot be told apart: each arrangement appears k! times, so you divide by k! (Monday’s fruit, the repeated letters of PALABRA). The typical mistake of dividing by 3 instead of 3!.
- The sum rule: a crossing is reached from the left or from below, never from both at once. It counts routes in any town, including closed streets.
- The product rule with routes: going through a crossing multiplies; going past one place or another (never both) adds. Counting by complement.
- Bijections: each row of jam jars is one of Ana’s routes and each route a row; that is why there are as many.
- Choosing k of n when order does not matter: $\binom{n}{k} = \frac{n!}{k!,(n-k)!}$ (triangles on 7 points, teams).
- Pascal’s triangle: Ana’s grid turned round; row n, position k (counting from 0).
- Combinatorial proofs without calculation: symmetry $\binom{n}{k} = \binom{n}{n-k}$ (choosing who goes is choosing who stays) and Pascal’s rule (Eugenio, who is late, splits the teams into two piles).
- A set of n elements has 2^n subsets, in two ways: light bulbs on or off, and the sum of a row of the triangle.
- Patterns: triangular numbers, the hockey stick, Fibonacci, powers of 11 and the Sierpinski triangle from the odd numbers.
- Newton’s binomial theorem: the coefficient of $a^k b^{n-k}$ in $(a+b)^n$ is $\binom{n}{k}$.
How it works
Students bet before calculating (how many routes are there?) and compare with the result. The first crossings are counted by drawing every route; when drawing them gets too long, the idea that solves it appears. Answers are marked instantly, with hints after mistakes and the solution after the fourth.
Materials
The slide decks «Combinatoria desde cero» I-III (in the lab’s materials, in Spanish) follow the same thread: Ana’s routes, the jars, Eugenio who is late, and classic bets.
Room-by-room contents
Room 1 · Coins and dice
Product rule with repetition: nᵏ. After the coins, a tree that doubles with each coin.
Student tasks
- 4 coins (16; 8 is the typical mistake).
- 3 dice (216), 2 coins and 1 die (24), a 4-digit PIN (10,000).
Room 2 · In a row: the factorial
Problem → idea → more problems. With 3 on a bench all the ways are found; with 4, pick the first and 3! remain.
Student tasks
- Find the 6 ways to seat 3.
- 4 on the bench (24); 5 books (120); a podium of 8 (336, not 8!); NUMEROS (5040).
Room 3 · Ana’s fruit
Things that cannot be told apart: each arrangement appears k! times, so divide by k!. Animation: Monday’s two fruits swap and merge.
Student tasks
- Monday to Friday, one a day (120).
- Two on Monday (60); 6 fruits (360); two on Monday and two on Tuesday (30).
Room 4 · Repeated letters
The same with letters: the A’s of PALABRA with subscripts, their 3! orders merge into one.
Student tasks
- MORENO (360) and PALABRA (840; divide by 3!, not by 3).
- With 4 equal letters, divide by what? (24).
Room 5 · A different route every day
Ana walks from home to school leaving a trail; 8 of her 20 routes appear and students bet how many there are.
Student tasks
- Bet (not marked).
- Draw a route; how many blocks she always walks (6); draw one that is not among the 8.
Room 6 · Start small
Crossings on the edges (only one route) and on 2nd Street, drawing every route.
Student tasks
- Bottom street and left avenue (1).
- All routes to (1, 1) and (2, 1) (2 and 3); to (3, 1) without drawing (4).
Room 7 · The sum rule
The 6 routes to crossing (2, 2) split by their last stretch: 3 from the left + 3 from below. The sum animated crossing by crossing up to the school.
Student tasks
- Find the 6 routes to (2, 2).
- (3, 2) = 10, (2, 3) = 10 and the school, 20 (compared with the bet).
Room 8 · Closed streets
The sum rule in other towns; a closed stretch brings 0.
Student tasks
- 5 × 2 (21) and 4 × 3 (35).
- With roadworks on 2nd Street (23) and with two closures (20).
Room 9 · Stopping at the bakery
Product rule with routes: 3 routes to the bakery × 3 from it, in a 3 × 3 table.
Student tasks
- To and from the bakery (3 and 3); through it (9; 6 is the adding mistake).
- Those that avoid it (11); bakery or kiosk (18); when do you multiply?
Room 10 · Two jars and two jars
2 jars of one jam and 2 of another: swap them until the 6 rows are found; with numbers, 4! = 24 divided by 2 · 2.
Student tasks
- Find the 6 rows.
- With numbers (24); how many numbered rows give the same row (4); different rows (6).
Room 11 · Three and three: divide by 3!
3 and 3 jars: each row appears 3! · 3! times.
Student tasks
- 720 with numbers; 6 when one kind loses its numbers (not 3); 36 when all do.
- Different rows (20; 80 is the mistake of dividing by 3 · 3).
Room 12 · Jars and routes are the same
One kind of jar for north, one for east: Ana walks and a jar drops for each block. Each route is a row and each row a route (bijection).
Student tasks
- From route to jars (multiple choice) and from jars to route (draw it).
- 4 + 2 jars in a 2 × 4 town (15); why there are as many routes as rows.
Room 13 · Triangles from seven points
Choosing k of n without order: with order, each triangle appears 3! times.
Student tasks
- Triangles (35; 210 with order) and segments (21).
- 4 points on a line (31); teams of 3 from 7 (35) and why it matches.
Room 14 · Pascal’s triangle
Ana’s town turns round and becomes the triangle. C(n, k) = routes = choosing k of n; counting starts at 0.
Student tasks
- Complete rows 4, 5 and 6.
- Tap C(6, 2); C(7, 3) = 35; C(10, 4) = 210 with the formula.
Room 15 · Why it is symmetric
Combinatorial proof: choosing the 2 guests is choosing the 3 who stay (the 10 pairs matched with their trios). The route reflected in the diagonal.
Student tasks
- Why C(5, 2) = C(5, 3).
- C(20, 18) = 190; what happens to a route when reflected.
Room 16 · Eugenio is late
Combinatorial proof of Pascal’s rule: the 35 teams of 3 from 7 split into those with Eugenio (15) and those without (20).
Student tasks
- Which two groups they split into.
- C(7, 3) = C(6, 3) + C(6, 2); the version with the last stretch of a route.
Room 17 · Marta’s friends
Subsets: 2ⁿ with light bulbs on or off, and the sum of a row of the triangle.
Student tasks
- The 7 groups of 3 friends.
- With 5 (31; 32 if the empty group counts) and with 6 (63).
Room 18 · Hidden treasures
Patterns in the triangle.
Student tasks
- Triangular numbers (21), the hockey stick (tap the 10), Fibonacci (34).
- Powers of 11 (14,641); the sum of a row is double the one before.
Room 19 · Even and odd: Sierpinski
The whole triangle painted; students unpaint the even numbers and Sierpinski appears, growing to 64 rows.
Student tasks
- Unpaint the even numbers in the first 8 rows.
- Which shape appears; odd numbers in 8 rows (27) and in 16 (81).
Room 20 · Newton’s binomial
(a + b)² as an area; (a + b)³ choosing a or b from each bracket, like Ana’s routes.
Student tasks
- Coefficients of (a + b)⁴ (1 4 6 4 1).
- a²b³ in (a + b)⁵ (10); x² in (x + 1)⁶ (15); sum of the coefficients of (a + b)⁷ (128).
Rooms to project
The most striking ones to show and discuss in class.