Pascal's Triangle
Generate rows of Pascal's triangle
About This Calculator
Pascal's triangle is a triangular array of numbers where each element is the sum of the two directly above it. It encodes binomial coefficients, powers of 11, Fibonacci numbers, and combinatorial identities all in one elegant structure.
Formula
Each element C(n,k) = C(n-1,k-1) + C(n-1,k)
Row n contains C(n,0), C(n,1), ..., C(n,n) = binomial coefficients
Sum of row n = 2^n
Example Calculation
Generate row 5 of Pascal's triangle
- C(5,0)=1, C(5,1)=5, C(5,2)=10, C(5,3)=10, C(5,4)=5, C(5,5)=1
- Row sum = 1+5+10+10+5+1 = 32 = 2^5
Row 5: 1, 5, 10, 10, 5, 1
Pascal's Triangle Rows 0-7
| Row | Values | 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 |
Frequently Asked Questions
How is Pascal's triangle used in the binomial theorem?
The coefficients in the expansion of (a+b)^n are exactly the values in row n of Pascal's triangle. For example, (a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3, matching row 3: 1, 3, 3, 1.
Where are Fibonacci numbers hidden in Pascal's triangle?
Sum the shallow diagonals of Pascal's triangle (going up-right at 45 degrees). The sums are 1, 1, 2, 3, 5, 8, 13, 21... — the Fibonacci sequence.
What pattern appears when you color odd numbers in Pascal's triangle?
Coloring only odd numbers creates Sierpinski's triangle, a famous fractal. This reveals a deep connection between Pascal's triangle and self-similar geometric patterns.