Pascal's Triangle Calculator

How to use the Pascal's triangle calculator

What this calculator does

This calculator generates exact rows of Pascal's triangle. Every value in row n is a binomial coefficient C(n,k), so the same output can be used to count combinations, supply coefficients for a binomial expansion, or inspect the distribution of equally likely binary outcomes. It calculates discrete mathematical identities; it does not decide whether a real-world probability model is appropriate or whether outcomes are independent.

When to use it

Use it to check homework on binomial coefficients, obtain the coefficients of (x + y)ⁿ, count unordered selections of k objects from n, or compare the relative counts of outcomes in repeated two-outcome experiments. The NIST Digital Library of Mathematical Functions treatment of binomial coefficients gives the formal definition, recurrence, symmetry, and generating-function identities behind these results.

How to calculate

1. The calculator opens with a complete demonstration: Show is set to “Triangle through row,” and Row number (n) is 10. The results and a validated example Excel workbook are immediately available.
2. Choose Triangle through row to display every row from 0 through the selected row, or choose Single row when you only need the coefficients for one power or one combinations problem.
3. Replace Row number (n) with a nonnegative whole number. Results update live. Read the selected row, summary cards, triangle, and coefficient table from the same exact model.
4. Select Download Excel to export the current inputs, summaries, and coefficient data as a real .xlsx workbook. Select Reset to clear the demonstration data. Reset leaves the calculator empty and disables export until a complete valid row number is entered again.

Input guide

Show is a required selection. “Triangle through row” produces rows 0 through n and accepts n from 0 to 50; for example, 10 displays eleven rows. “Single row” produces only row n and accepts values up to 150; for example, 20 returns 21 coefficients. Switching to a full triangle increases the amount of displayed and exported data. A common mistake is expecting “row 1” to be the top value; this calculator follows the standard zero-based convention, so the top is row 0.

Row number (n) is required and accepts digits for one whole number only – no commas, decimal points, signs, spaces inside the number, or scientific notation. The minimum is 0. Entering 10 generates row 10, while increasing the value adds one coefficient per row and rapidly increases the size of central coefficients. Fractions such as 5.5 are invalid because Pascal's triangle rows are indexed by nonnegative integers.

Output guide

Selected row identifies the active zero-based row. Row values lists the exact coefficients from C(n,0) through C(n,n). Number of entries is exactly n + 1. Row sum (2ⁿ) is the exact sum of all coefficients. Largest coefficient is the central value, or either of the equal central values when n is odd. Symmetric row confirms the identity C(n,k) = C(n,n-k).

The Pascal triangle block shows all rows through n in triangle mode and highlights the selected final row. In single-row mode it shows only the requested row. The Selected row detail table lists Position k, the exact Coefficient C(n,k), the matching Mirror position n-k, and each coefficient's Share of row sum. A zero row number is valid: it returns one coefficient, one entry, and a row sum of one.

Worked example

With the startup value n = 10, the calculator returns 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1. There are 10 + 1 = 11 entries. The row sum is 2¹⁰ = 1,024. The largest coefficient is C(10,5) = 10!/(5!5!) = 252. In a binomial expansion, these are the coefficients of (x + y)¹⁰; in a combinations problem, 252 is the number of ways to choose 5 objects from 10 without regard to order.