The general term

The term with k = r in Σ nCk aⁿ⁻ᵏ bᵏ is nCr aⁿ⁻ʳ bʳ. Counting terms from 1, it is the (r + 1)th term: Tᵣ₊₁ = nCr aⁿ⁻ʳ bʳ.

Any single term can be written without the full expansion. In (x + 2)⁶ the 4th term has r = 3: 6C3 x³ 2³ = 20 × 8 x³ = 160x³, which matches the full expansion.

To find the coefficient of a given power, write the power of x in Tᵣ₊₁ in terms of r and solve for r. In (2x + 1)⁷ the term in x⁴ needs 7 − r = 4, so r = 3 and the coefficient is 7C3 × 2⁴ = 35 × 16 = 560.

Always include the number carried by each part of the binomial: a = 2x brings 2 raised to the power of x, and b = −y brings a sign (−1)ʳ.

When a = x² and b = 2/x the powers of x combine: in (x² + 2/x)⁶, Tᵣ₊₁ = 6Cr 2ʳ x¹²⁻³ʳ. The term free of x has 12 − 3r = 0, so r = 4 and it equals 6C4 × 2⁴ = 15 × 16 = 240.

The largest binomial coefficient of row n sits in the middle: nC(n/2) for even n (for n = 6 it is 20), and the two equal middle ones for odd n.