Last updated September 24, 2026 · Report an error
Pascal's triangle is a triangular array of binomial coefficients, where each entry is simply the sum of the two entries diagonally above it, with a 1 anchoring both edges of every row. Named after Blaise Pascal (though the pattern was documented by earlier mathematicians in India, Persia, and China centuries before him), it's a compact way to read off the coefficients of any binomial expansion (a+b)ⁿ, and it also encodes combinatorics directly — the k-th entry of row n equals C(n,k), the number of ways to choose k items from a set of n. It shows up in probability (binomial distribution probabilities), combinatorics coursework, and even in patterns related to fractals when coefficients are colored by parity (the Sierpiński triangle).
How it works
Enter the row number n (rows are numbered starting from 0). The calculator computes C(n, 0) through C(n, n) using the multiplicative binomial coefficient formula, C(n,k) = n!/(k!(n-k)!), and can optionally list every row from 0 up to n, letting you see the full triangle build up.
- Enter row number (n).
- Enter also show rows 0 through n.
- Click Calculate to see your results.
Examples
Row 4
[1, 4, 6, 4, 1] — the coefficients of (a+b)⁴, since C(4,0)=1, C(4,1)=4, C(4,2)=6, C(4,3)=4, C(4,4)=1.
Row 6
[1, 6, 15, 20, 15, 6, 1] — notice each interior entry is the sum of the two entries above it in row 5 ([1,5,10,10,5,1]), e.g. 15=5+10.
Common mistakes to avoid
- Forgetting that row numbering starts at 0, so "row 4" has 5 entries, not 4.
- Assuming the entries in row n are just n repeated — they follow the binomial coefficient pattern, not a simple linear sequence.
- Miscounting when manually summing two entries from the row above to check a value — remember to align diagonally, not directly above.