Grow the Triangle
Every entry is a binomial coefficient ⁿCᵣ. Drag the slider to grow the triangle, then hover any tile to see the two parent tiles above it add together to build it — the rule Pascal noticed by hand.
Build (a + b)ⁿ Term by Term
Type any two quantities and a power. Each tile below is one term ⁿCᵣ · an−r · br — notice the power of a always falls and the power of b always rises, while the two powers always add up to n.
(x − y)ⁿ — signs alternate
Setting b = −y flips the sign of every other term. The size of each coefficient never changes — only the sign does.
(1 + x)ⁿ at x = 1 → 2ⁿ = ⁿC₀ + ⁿC₁ + ... + ⁿCₙ
Every coefficient in the row is positive and they all add up exactly to 2ⁿ — the total count of subsets of an n-element set.
(1 − x)ⁿ at x = 1 → 0 = ⁿC₀ − ⁿC₁ + ⁿC₂ − ...
Now every other bar flips below the line. Positives and negatives are built from the very same coefficients, so they cancel to exactly 0 (for n ≥ 1).
Cracking (98)⁵ without long multiplication
Write 98 = 100 − 2, then expand (100 − 2)⁵ with the binomial theorem. Every term is easy arithmetic because powers of 100 just shift decimal places.
Which is bigger — a huge power, or a plain number?
(1.01)1,000,000 is too big to ever multiply out, but the binomial theorem only needs its first two (positive) terms to already beat a comparison number — every remaining term only adds more.
Why 6ⁿ − 5n always leaves remainder 1 (mod 25)
Write 6 = 1 + 5 and expand (1 + 5)ⁿ. The first two terms give exactly 1 + 5n; every other term carries a factor of 5² = 25, so subtracting 5n always leaves a clean multiple of 25, plus 1.