Pascal's Triangle

Binomial coefficients in Pascal's triangle

Rows 0 to N−1 (1–15)
C(4, 2) = 4! / (2! · 2!) = 6
Binomial coefficient
C(4, 2) = 6
Pascal's rule (sum of cells above)
3 + 3 = 6
Sum of row n
24 = 16
Pascal's TrianglePascal's triangle with 6 rows; cell C(4, 2) = 6 highlighted with its two parent cells11112113311464115101051

この計算機について

The Pascal's Triangle Calculator builds the triangle of binomial coefficients row by row and highlights any cell C(n, k) together with the two cells above it. It shows how Pascal's rule C(n, k) = C(n−1, k−1) + C(n−1, k) generates each entry, the binomial-coefficient formula n! / (k!(n−k)!), and the row sum 2ⁿ. Useful for combinatorics, the binomial theorem, and probability.

一般的な例

  • Row 4 is 1 4 6 4 1 — the coefficients of (a + b)⁴
  • C(4, 2) = 6 = C(3, 1) + C(3, 2) = 3 + 3 (Pascal's rule)
  • The edges are all 1: C(n, 0) = C(n, n) = 1
  • Each row sums to a power of two: row 4 sums to 16 = 2⁴
  • Every row reads the same forwards and backwards: C(n, k) = C(n, n−k)