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The binomial expansion

The binomial expansion multiplies out (a + b)n quickly, without expanding bracket after bracket.

Pascal's triangle

Each row starts and ends with 1, and every other number is the sum of the two above it. Row n gives the coefficients of (a + b)n:

(a + b)4: 1, 4, 6, 4, 1.

nCr notation

n! = n x (n - 1) x ... x 2 x 1. The coefficients are

nCr = n! / (r!(n - r)!), also written as (n r) in a bracket.

5C2 = 5!/(2! 3!) = 10.

The expansion

(a + b)n = an + nC1 an-1 b + nC2 an-2 b2 + ... + bn.

The term in br is nCr an-r br. Put brackets round terms like (2x) or (-3x) so powers and signs are applied to the whole term.

For (1 + x)n: 1 + nx + n(n - 1)/2! x2 + n(n - 1)(n - 2)/3! x3 + ...

Worked example

Expand (2 + x)4.

  1. Coefficients 1, 4, 6, 4, 1
  2. 1(24) + 4(23)x + 6(22)x2 + 4(2)x3 + x4

Answer: 16 + 32x + 24x2 + 8x3 + x4

Key idea: The term in xr of (a + bx)n is nCr an-r (bx)r. Bracket the whole term before raising it to a power.

Check yourself

  1. Evaluate 5C2.
    Show answer

    10. 5!/(2! x 3!) = 120/(2 x 6) = 10.

  2. Evaluate 8!/6!.
    Show answer

    56. 8!/6! = 8 x 7 = 56.

  3. Find the coefficient of x2 in the expansion of (1 + 3x)5.
    Show answer

    90. 5C2 (3x)2 = 10 x 9x2 = 90x2.

  4. Find the coefficient of x3 in the expansion of (2 - x)6.
    Show answer

    -160. 6C3 x 23 x (-x)3 = 20 x 8 x (-x3) = -160x3.

  5. Find the first three terms of (1 + 2x)4 in ascending powers of x.
    Show answer

    1 + 8x + 24x2. 1 + 4(2x) + 6(2x)2 = 1 + 8x + 24x2.

  6. Which row of Pascal's triangle gives the coefficients of (a + b)4?
    • 1, 4, 6, 4, 1
    • 1, 3, 3, 1
    • 1, 5, 10, 10, 5, 1
    • 1, 4, 4, 1
    Show answer

    1, 4, 6, 4, 1. (a + b)4 has 5 terms with coefficients 1, 4, 6, 4, 1.

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Last updated 7 November 2026. Spotted a mistake? Tell us.