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Pascal's Triangle & Binomial Expansion (with steps)

Generate Pascal rows, C(n,k), and (ax + by)n with BigInt precision while How it's calculated stays pinned beside the output for classroom walkthroughs.

The tool highlights any C(n,k) you choose, surfaces row identities such as 2n and the alternating sum, and exports CSV logs plus shareable URLs for lesson plans.

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Pascal's triangle grows row by row using the recurrence C(n,k) = C(n-1,k-1) + C(n-1,k), and this calculator reveals each step while keeping everything in BigInt. Beyond rows, it evaluates combinations directly, expands (ax + by)n, and verifies the classic identities involving 2n, alternating sums, and Lucas-derived odd counts.

Every run is designed for teaching: copy a highlight-ready URL, drop the CSV log into slides, or use the narrated How it's calculated pane to demonstrate the recurrence and binomial theorem live.

Inputs & mode

Optional: highlight C(n,k) when viewing a row.

Result

How it's calculated

    Frequently asked questions

    What limits apply to n and the expansion?

    Rows are generated up to n = 200 for stability, and the binomial expansion is capped at n = 20 so coefficients remain readable. If you enter a larger value, the calculator shows a warning and safely clamps to those limits.

    What appears in How it's calculated?

    Each run narrates the Pascal recurrence, the iterative C(n,k) formula, the binomial theorem, and the proofs of the row identities, so you can follow every step and export the log as CSV for lessons.