Pascal's triangle starts with a 1 at the top. Every row begins and ends with 1, and each number in between is the sum of the two numbers directly above it. That one rule produces a grid packed with patterns: row sums that double, the counting numbers, the Fibonacci sequence, and the exact coefficients you need to expand (a + b)ⁿ or work out coin-flip odds.
Write a 1. Under it, write 1 1. For every row after that, put a 1 at each end and fill the middle by adding pairs from the row above. Row 3 is 1 3 3 1, so row 4 is 1, then 1 + 3 = 4, then 3 + 3 = 6, then 3 + 1 = 4, then 1. That gives 1 4 6 4 1.
One convention to know up front: the top row is called row 0, not row 1. It sounds fussy, but it makes the math line up. Row 4 is the row that expands (a + b)⁴, and its second number is 4.
| Row (n) | Numbers | Row sum (2ⁿ) |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 1 | 2 |
| 2 | 1 2 1 | 4 |
| 3 | 1 3 3 1 | 8 |
| 4 | 1 4 6 4 1 | 16 |
| 5 | 1 5 10 10 5 1 | 32 |
| 6 | 1 6 15 20 15 6 1 | 64 |
| 7 | 1 7 21 35 35 21 7 1 | 128 |
| 8 | 1 8 28 56 70 56 28 8 1 | 256 |
| 9 | 1 9 36 84 126 126 84 36 9 1 | 512 |
| 10 | 1 10 45 120 210 252 210 120 45 10 1 | 1,024 |
Look at the last column. Each row adds up to exactly double the row before, which is 2ⁿ. That's because every number gets used twice when you build the next row, once for the neighbor on each side.
Each entry is a combination: the number of ways to choose k items from n. Mathematicians write it as C(n, k) or "n choose k." Row 5 is 1 5 10 10 5 1, and the 10 in position 2 (counting from 0) tells you there are 10 ways to pick 2 things out of 5.
That's why the triangle shows up anywhere you're counting choices. How many different 3-topping pizzas can you make from 8 toppings? Go to row 8, position 3: 56. How many ways can 2 people out of 10 be chosen for a committee? Row 10, position 2: 45. You can use the formula n! ÷ (k! × (n − k)!), but for small numbers, reading it off the triangle is quicker.
The triangle is also symmetrical. Choosing 3 toppings to include is the same as choosing 5 to leave out, so row 8 reads the same forwards and backwards.
The diagonals are where it gets fun. Read down each slanting line from the edge and you get a different famous sequence:
| Diagonal | Sequence | What it counts |
|---|---|---|
| 1st | 1, 1, 1, 1, 1 … | Just the edges |
| 2nd | 1, 2, 3, 4, 5 … | Counting numbers |
| 3rd | 1, 3, 6, 10, 15 … | Triangular numbers (dots in a triangle) |
| 4th | 1, 4, 10, 20, 35 … | Tetrahedral numbers (balls stacked in a pyramid) |
| Shallow diagonals, summed | 1, 1, 2, 3, 5, 8 … | Fibonacci numbers |
A few more worth knowing:
This is the use most students meet first. When you expand (a + b)ⁿ, the coefficients are row n of the triangle. The powers of a count down from n while the powers of b count up from 0.
For (a + b)⁴, take row 4 (1 4 6 4 1) and you get:
a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
It saves a lot of multiplying. Expanding (x + 2)³ the long way takes three lines of work. With row 3 (1 3 3 1), it's x³ + 3·x²·2 + 3·x·4 + 8, which is x³ + 6x² + 12x + 8. That shortcut is known as the binomial theorem, and it's the same idea spreadsheets and stats software use under the hood.
Flip a fair coin n times and row n tells you how many ways each result can happen. Divide by the row sum and you've got the probability. With 4 flips, row 4 says there are 6 ways to get exactly 2 heads out of 16 equally likely outcomes, so the chance is 6/16, or 37.5%.
Row 10 works the same way for ten flips:
| Heads in 10 flips | Ways (row 10 entry) | Probability |
|---|---|---|
| 0 or 10 | 1 | 0.10% each |
| 1 or 9 | 10 | 0.98% each |
| 2 or 8 | 45 | 4.39% each |
| 3 or 7 | 120 | 11.72% each |
| 4 or 6 | 210 | 20.51% each |
| 5 | 252 | 24.61% |
| Total | 1,024 | 100% |
So even the most likely result, five heads and five tails, only happens about a quarter of the time. People expect a fair coin to split evenly, but in ten flips you'll miss 5-5 roughly three times out of four. If you want to test that yourself, run a batch of flips on the coin flip tool, or use the probability calculator for other odds.
Not Pascal, at least not first. The Indian scholar Pingala worked with the same combinations for poetic meter well over 2,000 years ago. Around the year 1000, Al-Karaji described it in Persia, and Omar Khayyam later used it. In China it's known as Yang Hui's triangle after the 13th-century mathematician who published it. Blaise Pascal wrote his treatise on it in 1654 and tied it firmly to probability, which is why his name stuck in Europe.
Add the two numbers above to get the one below, and remember that the top row is row 0. From there, row n gives you the coefficients for (a + b)ⁿ, the number of ways to choose k items from n, and the odds for n coin flips. Once you know where to look, it's one of the handiest tables in math.
Build Pascal's triangle to as many rows as you need and read off any value.
Pascal's Triangle Generator →Start with 1 at the top. Each row begins and ends with 1, and every number in between is the sum of the two numbers directly above it. Row 3 is 1 3 3 1, so row 4 is 1 4 6 4 1.
Row n adds up to 2 to the power of n. Row 4 sums to 16 and row 10 sums to 1,024. Each row doubles the one before it.
The numbers in row n are the coefficients of (a + b)ⁿ. Row 4 is 1 4 6 4 1, so (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.
Add the numbers along the shallow diagonals, the lines that go up one row and across two positions. Their sums are 1, 1, 2, 3, 5, 8, 13 and so on.
Numbering from 0 makes row n line up with (a + b)ⁿ and with n choose k. The top 1 is (a + b)⁰, so it's row 0.