Pascal’s Triangle Row Calculator – Any Row n

Each Pascal row lists the binomial coefficients for (a+b)ⁿ, and any row builds additively from the one above — no factorials needed. Row 10 emerges from row 9 by neighbor addition: 10 = 1+9, 45 = 9+36, 120 = 36+84, 210 = 84+126, and the central 252 = 126+126, then the row mirrors back down to 1. The calculator prints the target row, the construction from the previous row, and individual entries on demand — C(10,5) = 252, the middle of the symmetry. Row facts stream alongside: the sum is 2¹⁰ = 1024, the alternating sum is 0, and the second entry always equals n (so row 10 starts 1, 10). Large n is handled with exact integers — row 50 peaks at C(50,25) ≈ 1.26×10¹⁴ — and the hockey-stick identity (1+3+6+10 = 20) is available for cumulative sums down a diagonal.

1. 1

   Enter the row number n

   Row 0 is the apex 1

2. 2

   Build or extract

   The whole row or a single entry C(n,k)

3. 3

   See the construction

   Neighbor additions from row n−1 are listed

4. 4

   Verify identities

   Row sum 2ⁿ and symmetry checks included

• 1

  Row n has n+1 entries and sums to 2ⁿ

• 2

  Second entry always equals n

• 3

  Symmetry: C(n,k) = C(n,n−k) halves the computing work

• Counting the apex as row 1 — it is row 0

• Off-by-one in entry position k (0-based)

• Doubling the central entry of even rows by symmetry confusion

• Overflowing with floats where exact integers work fine