Pascal's Triangle | Moosa Academy

Pascal's Triangle is a triangular arrangement of numbers built from one rule so simple you can extend it on paper without a calculator: each inner number is the sum of the two directly above it. Several civilisations discovered it independently long before Pascal, yet it carries his name — and it turns up in the binomial theorem, in combinations, and throughout probability.

Concept The triangle itself

Every row is generated from the row above it. Nothing else is needed — no formula, no lookup.

Concept How the triangle is built

Each inner number = the number above left + the number above right.

In the row

1,\ 4,\ 6,\ 4,\ 1

, the neighbours 4 and 6 sit above.

4 + 6 = 10

, so 10 appears between them in the next row.

That is how

1,\ 5,\ 10,\ 10,\ 5,\ 1

is produced.

Note Three patterns to notice

1. Every row begins and ends with 1. The outer positions have only one number above them, so nothing is added and the 1 simply carries down.

2. Each row holds one more entry than the row before. The first row has a single 1, the second has $1,\ 1$ , the third has $1,\ 2,\ 1$ — which is exactly what gives the arrangement its triangular shape.

3. Every row is symmetric about its middle. Reading a row forwards or backwards gives the same sequence:

1,\ 6,\ 15,\ 20,\ 15,\ 6,\ 1

, the 6 on the left matches the 6 on the right.

The two 15s mirror each other, with 20 alone at the centre.

Example Building the next row

The last row shown is $1,\ 6,\ 15,\ 20,\ 15,\ 6,\ 1$ . Write the row that follows it.

Start with 1, since every row opens with 1.

1 + 6 = 7

, then

6 + 15 = 21

, then

15 + 20 = 35

20 + 15 = 35

, then

15 + 6 = 21

, then

6 + 1 = 7

Close with 1.

⟹ 1, 7, 21, 35, 35, 21, 7, 1

The new row has eight entries, one more than the seven above it, and it reads the same in both directions — both patterns hold, as expected.

Note Where the triangle is used

The binomial theorem — the entries of a row are exactly the coefficients in the expansion of

(a+b)^n

Combinations — each entry counts the ways of choosing a number of items from a group.

Probability — those same counts give the likelihoods of outcomes such as repeated coin tosses.

Number patterns — the triangle hides many further sequences within its diagonals.

The row $1,\ 4,\ 6,\ 4,\ 1$ , for instance, gives the coefficients of $(a+b)^4$ directly — no multiplying out required.

Note Mistakes to avoid

Multiplying the two numbers above instead of adding them.

Forgetting to start and finish each row with 1.

Adding numbers that are not directly above the position being filled.

Producing a row with the same number of entries as the one before it.

Breaking the symmetry — if a row does not read the same backwards, an arithmetic slip has crept in.

Summary

Pascal's Triangle is built by one rule: each inner number is the sum of the two directly above it.
Every row starts and ends with 1.
Each row contains one more entry than the row before, which creates the triangular shape.
Every row is symmetric about its centre, which doubles as a check on your arithmetic.
The rows give the binomial coefficients and the values used in combinations and probability.