1 · IB Maths
Binomial theorem
Pascal’s triangle is the expansion of (x + y)ⁿ written as pure numbers. Pick a row and a term, then add coefficients a and b to build the term of (ax + by)ⁿ.
Predict first
Start from the default power n = 5. If you increase n by one to n = 6, what happens to the row sum 2ⁿ?
Each row sums to 2ⁿ, so one more power multiplies the sum by 2: 2⁵ = 32 becomes 2⁶ = 64.
Check it in the model: Move the Power n slider from 5 to 6 and read the Row sum 2ⁿ readout.
Click any entry to select it. Gold: chosen term. Green: the two entries above that add to it. Outlined: the symmetric partner.
general term: ⁿCₖ aⁿ⁻ᵏ bᵏ xⁿ⁻ᵏ yᵏ
ⁿCₖ = ⁿ⁻¹Cₖ₋₁ + ⁿ⁻¹Cₖ ; ⁿCₖ = ⁿCₙ₋ₖ
row sum: 2ⁿ (a = b = 1)
- Expansion of
- Coefficient ⁿCₖ
- Working
- Term k
- Pascal’s rule
- Row n
- Row sum 2ⁿ
Try this
- Set n = 6 and step k from 0 to 6. Check that the coefficients read 1, 6, 15, 20, 15, 6, 1, and that the row sum is 2⁶.
- Choose n = 8 and k = 3. Which entry is outlined as the partner, and why does it have the same value?
- Set n = 7, a = 2, b = −3 and k = 3. Verify the term of (2x − 3y)⁷ by hand from ⁷C₃ × 2⁴ × (−3)³.
Common error: forgetting that a and b are raised to the powers too, so the term of (2x − 3y)⁷ in y³ is ⁷C₃ × 2⁴ × (−3)³ and not 35 × 2 × (−3), and dropping the sign of an odd power of a negative number.
More on 1: Arithmetic and geometric series · Compound interest and depreciation · Proof by induction · Permutations and combinations · De Moivre's theorem · Systems of linear equations · Complex numbers on the Argand plane · Roots of a complex number · Complex loci · Matrix transformations · Sequences and series · Exponentials and logarithms