Pascal's Triangle
Pascal's triangle to any row, with an entry C(n, k) worked out, the row's sum, and the binomial expansion of (a + b)^n. Each number is the sum of the two above it.
C(6, 2)
- Row 6 7 entries, the same read from either end.
- 1, 6, 15, 20, 15, 6, 1
- Entry 2 of row 6 C(6, 2) = 6! / (2! x 4!) = 720 / (2 x 24) = 15. Also written nCr or "6 choose 2".
- 15
- From the row above Entries 1 and 2 of row 5, the two numbers above it.
- 5 + 10 = 15
- Its mirror
- C(6, 4) = 15
- Sum of row 6 2^6.
- 64
- Middle entry C(6, 3), the largest in the row.
- 20
- (x + y)^6 Row 6 is the list of coefficients of the expansion.
- x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6
| Row | Entries | Sum |
|---|---|---|
| 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 | 1024 |
How the Pascal's Triangle works
Pascal's triangle is a triangular array of whole numbers in which every entry is the sum of the two entries directly above it. It starts with a single 1 at the top, called row 0, and each row begins and ends with 1. Row 1 is 1 1, row 2 is 1 2 1, row 3 is 1 3 3 1, row 4 is 1 4 6 4 1, row 5 is 1 5 10 10 5 1, and row 6 is 1 6 15 20 15 6 1.
To build row 7, start and end with 1 and add each neighboring pair of row 6: 1 + 6 = 7, 6 + 15 = 21, 15 + 20 = 35, and so on, giving 1 7 21 35 35 21 7 1.
The entries are binomial coefficients. Entry k of row n, counting both from 0, is C(n, k) = n! / (k! (n - k)!), read "n choose k": the number of ways to choose k things from n. Entry 2 of row 6 is 6! / (2! x 4!) = 720 / 48 = 15, so there are 15 ways to pick 2 people from 6. The addition rule and the factorial formula always agree, because choosing k from n either includes a particular item (C(n - 1, k - 1) ways) or does not (C(n - 1, k) ways).
The binomial expansion. Row n lists the coefficients of (a + b)^n. Row 4 is 1 4 6 4 1, so (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. With other terms, each coefficient is multiplied by the powers of those terms: (2x - 3)^4 gives 16x^4 - 96x^3 + 216x^2 - 216x + 81.
Patterns in the triangle. Each row is symmetric. Row n sums to 2^n, so row 6 sums to 64. The diagonals running down from the edge are the 1s, then the counting numbers 1, 2, 3, 4, then the triangular numbers 1, 3, 6, 10, 15, then the tetrahedral numbers. Shading the odd entries draws the Sierpinski triangle. When n is prime, every entry of row n except the two 1s is a multiple of n, as row 7 (7, 21, 35) shows.
Big rows. As a Pascal triangle calculator, this one tabulates rows 0 to 20 and gives any single row or entry up to row 1,000, every entry an exact whole number. Entries grow fast: row 30's middle entry is 155,117,520, and row 1,000's has 300 digits.
Pascal's Triangle FAQ
How do you make Pascal's triangle?
Write 1 at the top as row 0. Each row after it starts and ends with 1, and every other entry is the sum of the two entries above it. Row 4 is 1 4 6 4 1, so row 5 is 1, 1 + 4, 4 + 6, 6 + 4, 4 + 1, 1, which is 1 5 10 10 5 1.
What is the formula for an entry in Pascal's triangle?
Entry k of row n is C(n, k) = n! / (k! (n - k)!), with both counted from 0. Entry 2 of row 6 is 720 / (2 x 24) = 15.
What does Pascal's triangle have to do with the binomial theorem?
Row n gives the coefficients of (a + b)^n. (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, and row 3 is 1 3 3 1.
What is the sum of a row?
2^n. Row 0 sums to 1, row 1 to 2, row 2 to 4, and row 10 to 1,024. Each row doubles the one above because every entry is added into the two entries below it.
Why does Pascal's triangle start at row 0?
So that row n is the coefficients of (a + b)^n and entry k is C(n, k). (a + b)^0 = 1, which is the single 1 at the top.