Row 10, entry 5 is 252: there are 252 ways to choose 5 items from 10. The numbers in row 15 add up to 32,768.
| Row n | k = 0 | k = 1 | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k = 7 | k = 8 | k = 9 | k = 10 | k = 11 | k = 12 | k = 13 | k = 14 | k = 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | |||||||
| 9 | 1 | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | ||||||
| 10 | 1 | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | |||||
| 11 | 1 | 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | ||||
| 12 | 1 | 12 | 66 | 220 | 495 | 792 | 924 | 792 | 495 | 220 | 66 | 12 | 1 | |||
| 13 | 1 | 13 | 78 | 286 | 715 | 1287 | 1716 | 1716 | 1287 | 715 | 286 | 78 | 13 | 1 | ||
| 14 | 1 | 14 | 91 | 364 | 1001 | 2002 | 3003 | 3432 | 3003 | 2002 | 1001 | 364 | 91 | 14 | 1 | |
| 15 | 1 | 15 | 105 | 455 | 1365 | 3003 | 5005 | 6435 | 6435 | 5005 | 3003 | 1365 | 455 | 105 | 15 | 1 |
How it is computed
Every row starts and ends with 1, and each inner number is the sum of the two above it: C(n, k) = C(n − 1, k − 1) + C(n − 1, k). The entry in row n, column k equals n! ÷ (k! (n − k)!). Row n sums to 2ⁿ.
FAQ
What is Pascal's triangle used for?+
Counting combinations, expanding (a + b)ⁿ (row 4 gives a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴) and computing coin-flip probabilities.
Why does the table look like a staircase?+
Row n has n + 1 entries, so the table is the triangle aligned to the left. The diagram shows its usual centred shape.