In this article
Pascal's triangle is built with one rule: every number is the sum of the two numbers directly above it, and the edges are all 1. It is named after the French mathematician Blaise Pascal, but it was known in India, Persia and China centuries earlier.
The first nine rows
| Row | Numbers |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
| 6 | 1 6 15 20 15 6 1 |
| 7 | 1 7 21 35 35 21 7 1 |
| 8 | 1 8 28 56 70 56 28 8 1 |
To get the 6 in row 4, add the 3 and 3 above it. To get the 56 in row 8, add 21 and 35 from row 7.
- 1Start with row 41 4 6 4 1
- 2Put a 1 at each end1 _ _ _ _ 1
- 3Add neighbours in pairs1+4 = 5, 4+6 = 10, 6+4 = 10, 4+1 = 5
- 4Row 5 is ready1 5 10 10 5 1
No multiplication is needed. Repeated addition produces every row.
Pattern 1: each row sums to a power of 2
Add up any row and you get 2 raised to the row number. Row 4: 1 + 4 + 6 + 4 + 1 = 16 = 2⁴. The reason is simple: each number feeds two numbers below it, so each row doubles the one before.
A perfect doubling staircase. Row 10 adds up to 1,024.
Pattern 2: the numbers are combinations
The number in row n, position k (both counted from 0) is C(n, k), the count of ways to choose k items from n without caring about order. In row 5, the 10 is C(5, 2): there are 10 ways to pick 2 people from 5.
| n \ k | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| n = 2 | 1 | 2 | 1 | |||
| n = 3 | 1 | 3 | 3 | 1 | ||
| n = 4 | 1 | 4 | 6 | 4 | 1 | |
| n = 5 | 1 | 5 | 10 | 10 | 5 | 1 |
| n = 6 | 1 | 6 | 15 | 20 | 15 | 6 |
| n = 7 | 1 | 7 | 21 | 35 | 35 | 21 |
The ringed cell is C(5, 2) = 10. Darker means a bigger count. Blank cells are outside the triangle.
Real examples: choosing a 5-card hand from 52 cards has C(52, 5) = 2,598,960 possibilities, and an 6-from-49 lottery has C(49, 6) = 13,983,816 possible tickets. The triangle gets there by repeated addition, and a formula does it in one step.
Pattern 3: coin flips
Flip a fair coin four times. There are 2⁴ = 16 equally likely sequences, and the number of those with exactly k heads is C(4, k): row 4 of the triangle.
Two heads and two tails is the most common result, with 6 of 16 sequences (37.5%). All heads has just 1 of 16 (6.25%).
Move to ten flips and row 10 gives the counts out of 1,024. The bell shape appears: results near 5 heads are far more common than extremes.
Exactly 5 heads happens 252 times out of 1,024, about 24.6%. All heads or all tails happens 2 times out of 1,024, about 0.2%.
Add the middle three counts, 210 + 252 + 210, to get 672. Even a fair coin gives lopsided results more often than most people expect.
Pattern 4: expanding (a + b)ⁿ
Multiply (a + b) by itself n times and the numbers in front of each term are row n of the triangle. For (a + b)⁴ the coefficients are 1, 4, 6, 4, 1.
| Power | Expansion |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a + b)³ | a³ + 3a²b + 3ab² + b³ |
| (a + b)⁴ | a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ |
| (a + b)⁵ | a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵ |
Set a = b = 1 and every coefficient adds to 2ⁿ, which is the row-sum pattern again.
Pattern 5: hidden Fibonacci numbers
Draw shallow diagonals through the triangle and add the numbers on each one. The sums are 1, 1, 2, 3, 5, 8, 13: the Fibonacci sequence. For example, 1 + 6 + 5 + 1 = 13 on one such diagonal.
- Sum 11A single 1.
- Sum 21 + 1 = 2Two 1s from neighbouring rows.
- Sum 51 + 3 + 1 = 5Picked from rows 2, 3 and 4.
- Sum 81 + 4 + 3 = 8Picked from rows 3, 4 and 5.
- Sum 131 + 5 + 6 + 1 = 13Picked from rows 3, 4, 5 and 6.
Fibonacci numbers: each is the sum of the two before it. The same numbers sit hidden inside the triangle.
More patterns to spot
From row 5 the entries reach 10 or more and carry over to the next place: row 5 (1, 5, 10, 10, 5, 1) becomes 161,051, which is 11⁵.
Questions people ask
What is Pascal's triangle?
It is a triangle of numbers where each number is the sum of the two directly above it, with 1s along both edges. Its rows contain the binomial coefficients.
How do you find a number in Pascal's triangle?
Add the two numbers above it. Or use the formula C(n, k) = n! ÷ (k! × (n − k)!) for the number in row n, position k, counting both from zero.
What do the row sums of Pascal's triangle equal?
Each row sums to 2 raised to the row number. Row 5 adds up to 32 and row 10 adds up to 1,024.
How is Pascal's triangle used in probability?
Row n gives the number of ways to get k successes in n fair yes-or-no trials, such as k heads in n coin flips. Divide each entry by 2ⁿ to get its probability.
Where is the Fibonacci sequence in Pascal's triangle?
Add the numbers along the shallow diagonals. The sums are 1, 1, 2, 3, 5, 8, 13 and so on.
Who invented Pascal's triangle?
Blaise Pascal wrote about it in the 1650s, but earlier mathematicians in India, Persia and China studied it long before. In China it is often called Yang Hui's triangle.