How is Pascal’s triangle built?

Each row begins and ends with 1, and every entry in between is the sum of the two entries above it — the identity C(n,k) = C(n−1,k−1) + C(n−1,k).

How does it relate to the binomial theorem?

Row n holds the coefficients of (a+b)ⁿ. Row 5 is 1 5 10 10 5 1, which gives a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵.

What does each row add up to?

Row n sums to 2ⁿ (put a = b = 1 into (a+b)ⁿ). Adding with alternating signs gives 0 instead (a = 1, b = −1).

What patterns are in the diagonals?

The first diagonal is all ones, the second the natural numbers, the third the triangular numbers 1, 3, 6, 10…, the fourth the tetrahedral numbers. Sums along the shallow diagonals produce the Fibonacci sequence.

Why is it also called Yang Hui’s triangle?

The Chinese mathematician Yang Hui recorded it in 1261, crediting the earlier Jia Xian. In the West it carries Pascal’s name after his systematic study of it in 1654.

🔺

🔺Pascal’s Triangle Generator - Binomial Coefficients and Expansions Online

Generate Pascal’s triangle and pick any row to see the matching (a+b)ⁿ expansion and its coefficients. A separate C(n, k) box handles large values of n, and everything is computed with big integers so the numbers stay exact.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

Loading tool...

How to use Pascal’s Triangle

1Choose how many rows to display (up to 30)
2Choose a row to highlight — its coefficients and binomial expansion appear below
3Use the C(n, k) box for a single binomial coefficient

FAQ

How is Pascal’s triangle built?: Each row begins and ends with 1, and every entry in between is the sum of the two entries above it — the identity C(n,k) = C(n−1,k−1) + C(n−1,k).
How does it relate to the binomial theorem?: Row n holds the coefficients of (a+b)ⁿ. Row 5 is 1 5 10 10 5 1, which gives a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵.
What does each row add up to?: Row n sums to 2ⁿ (put a = b = 1 into (a+b)ⁿ). Adding with alternating signs gives 0 instead (a = 1, b = −1).
What patterns are in the diagonals?: The first diagonal is all ones, the second the natural numbers, the third the triangular numbers 1, 3, 6, 10…, the fourth the tetrahedral numbers. Sums along the shallow diagonals produce the Fibonacci sequence.
Why is it also called Yang Hui’s triangle?: The Chinese mathematician Yang Hui recorded it in 1261, crediting the earlier Jia Xian. In the West it carries Pascal’s name after his systematic study of it in 1654.