Recall the problem from Lesson 4: with three objects A, B and C, how many different subsets can we make?
There is 1 subset with no objects, 3 subsets with one object, 3 subsets with two objects, and 1 subset containing all three. So, grouped by size, the counts are:
This row is part of a much bigger pattern. Let’s build it.
Build Pascal’s triangle
Start with a single 1. Put 1s down both edges. Every number inside is the sum of the two numbers just above it.
What is the next row?
Reveal answer
1 6 15 20 15 6 1
Each entry is a “choose” number
Look at row 5:
1 5 10 10 5 1
These are exactly the numbers of ways to choose 0, 1, 2, 3, 4 or 5 objects from 5 objects.
Reading from left to right, row n gives “n choose 0”, “n choose 1”, …, “n choose n”.
There are 6 students. A committee of 2 is chosen. Use Pascal’s triangle to find the number of possible committees.
Reveal answer
Row 6 is:
1 6 15 20 15 6 1
The entry for choosing 2 is 15.
Why do neighbouring entries add?
Suppose we choose 2 people from a group of 5. Focus on Alice. Every possible pair belongs to exactly one of two cases.
B C D E
Choose 1 more person from the remaining 4.
B C D E
Alice is left out, so both people come from the remaining 4.
So the number of pairs from 5 is the sum of the two neighbouring entries above it.
You do not need to memorise this formula. The important idea is “chosen / not chosen”.
Seven children are available and we want a team of 3. Split the teams into those containing Ben and those not containing Ben.
Reveal answer
With Ben: choose 2 from the other 6 → 15.
Without Ben: choose 3 from the other 6 → 20.
15 + 20 = 35
It is enough for today to understand:
rows count selections → neighbouring entries add because of two cases
Come back later for the second half, where we explore symmetry, diagonals and powers of 2.
Pattern detective
Pascal’s triangle contains much more structure. Try to spot each pattern before opening the explanation.
🪞 Why is every row symmetric?
Choosing k objects to include is equivalent to choosing n−k objects to leave out. So “n choose k” equals “n choose n−k”.
🔢 What are the first two diagonals?
The outside diagonal is all 1s. The next diagonal is 1, 2, 3, 4, 5, … because choosing one object from n gives n choices.
🔺 Can you spot triangular numbers?
The next diagonal begins 1, 3, 6, 10, 15, …. These are called triangular numbers because the same numbers of dots can be arranged into growing triangles.
In Pascal’s triangle they appear for a counting reason too: the entries on this diagonal count the number of ways to choose 2 objects from a growing set.
We explore triangular numbers in more detail — including the dot pattern and the formula behind them — in Magical Numbers: Lesson 6 →
✨ Advanced link: why are they called binomial coefficients?
The same numbers that count selections also appear when we expand powers of a two-term expression — a binomial.
The coefficients are 1, 4, 6, 4, 1 — exactly row 4 of Pascal’s triangle. That is why the numbers “n choose k” are called binomial coefficients.
This is an advanced connection: you do not need it to use Pascal’s triangle for counting.
Why do whole rows add to powers of 2?
Take 5 objects. Every object has exactly two independent choices: in the subset or out of it.
2 × 2 × 2 × 2 × 2 = 25 = 32
Pascal’s row 5 counts those same 32 subsets, sorted by size:
Instead of looking at the size of the numbers, look only at whether each entry is odd or even. In the rows below, odd entries are coloured purple and even entries are left pale.
What structure can you see? Look for smaller triangles repeating inside the larger one. Then predict which entries in the next row will be odd before calculating the row.
Reveal the next row
The next row is 1, 8, 28, 56, 70, 56, 28, 8, 1.
Only the two outside 1s are odd. If you continue colouring odd and even entries, a repeating triangular pattern emerges. At larger scales it becomes the same kind of self-similar pattern known as the Sierpiński triangle, a beautiful fractal. We will explore fractals in a future Sumthing Else course.
Look at the diagonal
1, 3, 6, 10, 15, 21, …
Each term is a triangular number.
Predict the next two terms. Then explain why each new term is obtained by adding the next counting number.
Reveal answer
The next terms are 28 and 36.
When a new object is added, it can form a pair with every object that was already there. So the triangular numbers grow by 2, then 3, then 4, then 5, and so on.
What have we learned?
Each inside entry is the sum of the two above it.
Row n contains all the “n choose k” counts.
Neighbouring entries add because choices split into “chosen” and “not chosen” cases.
Symmetry, diagonals and row sums all have counting meanings.
Download the printable handout, exercises and answers for Pascal’s Triangle.
Find the resources →Next: Clever counting
Sometimes the quickest way to count something is not to count it directly at all.
In the final lesson we will use complements, cases, symmetry and double counting to solve problems that look much harder than they really are.
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