Math
Pascal's Triangle Calculator
Pascal's triangle starts with a 1 and builds each row by adding the two numbers above.
Draw as many rows as you need with exact big-integer entries, or jump straight to one entry C(n, k) and see it highlighted, with its factorial formula, Pascal's rule, and the row sum 2^n.
Rows 0 through n - 1. Up to 40.
Try an example
Result
Result
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- Row sum
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- Largest entry
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- Row
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
How to read the answer
The primary result is the entry you asked for or the full set of rows. Each entry is a binomial The number multiplying a variable, like the 3 in 3x.: row n, position k (both counting from 0) equals C(n, k), the number of ways to choose k items from n. Row sums double each time (2^n), the triangle is symmetric, and every interior entry is the sum of the two entries above it, so a wrong entry means the row above it is wrong too.
How Pascal's triangle is built
Row 0 is 1. Every later row starts and ends with 1, and each inside entry is the sum of the two entries diagonally above it. Row 4 is 1 4 6 4 1 because 1 + 3 = 4, 3 + 3 = 6, and 3 + 1 = 4 from row 3 (1 3 3 1).
Rows 0 to 6
| Row n | Entries | Sum = 2^n |
|---|---|---|
| 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 |
What the entries mean
- Combinations: C(n, k) counts the ways to choose k things from n. Row 5, entry 2 is 10 because there are 10 ways to pick 2 of 5.
- Binomial expansion: row n gives the coefficients of (a + b)^n. (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
- Probability: C(n, k) / 2^n is the chance of exactly k heads in n fair coin flips.
- Row sums double each time because every subset of n items either includes or excludes the new item.
Patterns hidden in the triangle
- The second diagonal is the counting numbers 1, 2, 3, 4, ...; the third is the triangular numbers 1, 3, 6, 10, ...
- Shallow diagonals sum to the Fibonacci numbers.
- Each row is symmetric: C(n, k) = C(n, n - k).
- In a row with a A whole number above 1 whose only factors are 1 and itself. number n, every inside entry is divisible by n.
How to use this calculator
- Choose Draw rows and pick how many rows to see, or choose Look up one entry and give n and k.
- Read the triangle (scroll sideways for wide rows), the highlighted entry, and the row sum.
- Open the work for the Multiply a whole number by every whole number below it, so 4 factorial is 24. calculation and Pascal's rule applied to your entry.
Common mistakes
- Rows and positions start at 0, not 1. The 1 at the top is row 0, and the left edge of every row is position 0.
- k cannot exceed n; C(n, k) is 0 outside the row.
- C(n, k) is not n × k or n / k; use the factorial formula or the triangle.
Worked examples
Ten rows
Rows 0 to 9
Result
1 9 36 84 126 126 84 36 9 1
Single row
Row 0 is just 1
Result
1
C(10, 3)
Entry 3 of row 10 is 120
Result
120
C(5, 2)
Ways to pick 2 of 5
Result
10
Edge entry
C(7, 0) = 1
Result
1
Large entry
C(60, 30) has 18 digits
Result
118,264,581,564,861,424
Out of the row
k bigger than n
Result
—
Frequently asked questions
What is Pascal's triangle?+
A triangular array of numbers where each entry is the sum of the two above it, with 1s down both edges. Row n contains the binomial coefficients C(n, 0) through C(n, n).
What is row 10 of Pascal's triangle?+
1 10 45 120 210 252 210 120 45 10 1. Its entries sum to 2^10 = 1,024.
How do I find a specific number in Pascal's triangle?+
Use C(n, k) = n! / (k! (n - k)!), where n is the row and k the position, both counted from 0. Or use the entry lookup mode to see it highlighted.
Why does each row sum to a power of 2?+
Row n lists the coefficients of (1 + 1)^n = 2^n. Equivalently, the entries count all subsets of an n-element set, of which there are 2^n.
How is Pascal's triangle used for binomial expansion?+
Row n gives the coefficients of (a + b)^n in order. For (x + 2)^3 use row 3: 1, 3, 3, 1, giving x^3 + 3x^2(2) + 3x(4) + 8 = x^3 + 6x^2 + 12x + 8.
Where are the Fibonacci numbers in Pascal's triangle?+
Add along the shallow diagonals (up and to the right): 1, 1, 2, 3, 5, 8, ... appear as those sums.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026