Pascal's Triangle Calculator
Generate up to 20 rows of Pascal's Triangle and display the nCr (combination) values for any selected row.
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How to use this calculator
Each entry is the sum of the two entries above it. Row n contains the binomial coefficients C(n,0), C(n,1), …, C(n,n).
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Enter the number of rows to generate (1–20).
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Enter the row number (0-indexed) for which you want the individual nCr values displayed.
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Each row of Pascal's Triangle is shown, with the selected row highlighted.
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The nCr (combination) values and row sum (2^n) are displayed below the triangle.
Frequently asked questions
What are the numbers in Pascal's Triangle?
Each number is a binomial coefficient C(n,k) = n!/(k!(n−k)!), representing the number of ways to choose k items from n without regard to order. Row 0 is {1}, row 1 is {1,1}, row 2 is {1,2,1}, etc. These same numbers appear as coefficients when expanding (a+b)^n.
What is the row sum pattern?
The sum of all entries in row n is 2^n. Row 0: 1 = 2^0. Row 3: 1+3+3+1 = 8 = 2^3. This makes sense because summing C(n,0)+C(n,1)+…+C(n,n) counts all subsets of an n-element set — which is 2^n by the power set formula.
Where do Fibonacci numbers hide in Pascal's Triangle?
Sum the diagonals running from upper-right to lower-left: 1; 1; 1+1=2; 1+2=3; 1+3+1=5; 1+4+3=8; … These are the Fibonacci numbers. This connection between Pascal's Triangle and Fibonacci numbers has no simple closed-form proof but follows from a combinatorial identity.
What does Pascal's Triangle have to do with probability?
Binomial probabilities use C(n,k): the probability of exactly k successes in n independent trials with success probability p is C(n,k)·p^k·(1−p)^(n−k). Flipping 4 coins: the chances of 0,1,2,3,4 heads are proportional to row 4 of Pascal's Triangle: 1,4,6,4,1 (out of 16 equally likely outcomes).
Pascal's Triangle — binomial coefficients and combinatorics
Construction and the binomial theorem
Pascal's Triangle is constructed by placing 1s on both edges of each row and making each interior entry the sum of the two entries directly above it. Row n corresponds to the coefficients of the expansion (a+b)^n. For example, (a+b)^4 = a^4 + 4a³b + 6a²b² + 4ab³ + b^4, matching row 4: 1,4,6,4,1. This is the Binomial Theorem, and Pascal's Triangle is its visual representation.
Hidden patterns in the triangle
Pascal's Triangle is rich with patterns: powers of 2 in row sums, Fibonacci numbers in diagonal sums, powers of 11 in row values (11^0=1, 11^1=11, 11^2=121, 11^3=1331), triangular and tetrahedral numbers in the diagonals, and the Sierpinski triangle in the parity pattern (odd entries). Each pattern reflects a deep combinatorial identity.
Historical context and Pascal's contribution
The triangle was known to Chinese mathematician Yang Hui in the 13th century (called Yang Hui's Triangle in China and Khayyam's Triangle in Iran, where it appeared in the 11th century). Blaise Pascal systematized its properties in his 1665 Traité du triangle arithmétique. Pascal's name stuck in Western mathematics, but the triangle is a truly international discovery spanning many cultures and centuries.
Learn more from an authoritative source:
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