Pascal's triangle grows row by row from C(n,k) = C(n-1,k-1) + C(n-1,k). This calculator shows each step with BigInt values. It also evaluates combinations, expands (ax + by)n, and verifies common row identities.
Each run is built for teaching. Copy a highlight-ready URL, drop the CSV log into slides, or use the narrated How it's calculated pane in class.
Results
How it's calculated
How to use Pascal's Triangle effectively
Choose one mode, enter n, and run the calculation to see exact BigInt values for rows, combinations, or binomial coefficients. The step log explains the recurrence and the coefficient pattern so you can reuse the result in class notes or homework.
How it works
Triangle mode builds row n from exact combinations C(n,k). Combination mode highlights one entry from the same recurrence, and binomial mode maps the row onto coefficients of (ax + by)^n. Every mode shows the intermediate reasoning in a math-first order.
When to use
Use this page when you want to verify a row of Pascal's Triangle, check a single combination, or expand a binomial without switching tools. It is especially useful for lessons on combinatorics, binomial theorem, and parity patterns.
Common mistakes to avoid
- Entering n above the safe display limits and expecting every coefficient to stay readable.
- Confusing the row index n with the position k when checking one entry C(n,k).
- Typing invalid variable symbols in binomial mode instead of short Latin names such as x and y.
- Copying rounded display text into another worksheet instead of using CSV or the exact listed coefficients.
Interpretation and worked example
If you enter n = 5, triangle mode returns 1, 5, 10, 10, 5, 1. The same row explains why (x + y)^5 expands with those coefficients, and choose mode lets you isolate C(5,2) = 10 without rebuilding the full proof by hand.
See also
Frequently asked questions
What limits apply to n and the expansion?
Rows are generated up to n = 200 for stability. Binomial expansion is capped at n = 20 so coefficients stay readable. If you enter a larger value, the calculator warns you and clamps to these limits.
What appears in How it's calculated?
Each run explains the Pascal recurrence, iterative C(n,k), and binomial theorem in order. You can follow every step and export the log as CSV for lessons.
Which mode should I choose first?
Use triangle mode for the whole row, choose mode for one exact combination C(n,k), and expand mode when you want coefficients for (ax + by)^n. All three modes share the same recurrence, so switching modes is mainly about the output format you need.
Why is binomial expansion limited to smaller n?
Expansion mode is capped at n = 20 so the coefficient list and step log stay readable on mobile and classroom screens. Row mode can go much higher because it only needs one BigInt row, not a full symbolic expansion.
How reliable are the displayed values?
Values are computed exactly with BigInt inside the browser, then formatted for display. They are suitable for teaching, homework checks, and combinatorics practice, but you should still verify notation and symbol choices before publishing the result elsewhere.
What to read in the results
Rows and combinations
In triangle mode, the row lists every coefficient for n. In choose mode, the highlighted entry is exactly C(n,k), which matches the same number in the full row and lets you move from a table view to a single combination proof.
Binomial expansion
Expand mode reuses the same coefficients and places them on descending powers of x and y. This is the quickest way to connect the triangle to the binomial theorem and check signs or coefficients before writing the final expression by hand.
Pattern checks
The row sum, alternating sum, and odd-coefficient count help you verify familiar identities such as 2^n and parity patterns. These summaries are useful when you want more than a list of coefficients and need a compact classroom talking point.
Export and sharing
Use CSV when you want a table for slides or worksheets, and use the share button when you want students to reopen the same n, k, or binomial inputs. This keeps the exact setup reproducible across devices.