Single correct · +4 / −1

In (2x + 3)⁹, the coefficient of x⁶ is:

  1. (a) C(9,6) × 2⁶ × 3³
  2. (b) C(9,3) × 2³ × 3⁶
  3. (c) C(9,6) × 2⁶ × 3⁶
  4. (d) C(9,6) × 2³ × 3³
Answer and solution

Answer (a) C(9,6) × 2⁶ × 3³

Solution

Maths sir · 4 steps

The power of 2 always follows the power of x, so for x⁶ you need (2x)⁶, which leaves 3³.

  1. General term. In (2x + 3)⁹, the term with 3 raised to the power r is written below. The powers of 2x and 3 always add up to 9.

    Tᵣ₊₁ = C(9, r)(2x)⁹⁻ʳ 3ʳ = C(9, r) 2⁹⁻ʳ 3ʳ x⁹⁻ʳ

  2. Match the power. For x⁶ you need 9 - r = 6, so r = 3. The coefficient is C(9, 3) × 2⁶ × 3³.
  3. Use symmetry. Since C(9, 3) = C(9, 6) = 84, this is the same as C(9, 6) × 2⁶ × 3³. Its value is 84 × 64 × 27 = 145152.
  4. So. The coefficient of x⁶ is C(9,6) × 2⁶ × 3³, which is option (a).

The idea to keep. In (ax + b)ⁿ, the power of a equals the power of x, and the powers of a and b add to n. Also remember that C(n, r) = C(n, n-r).

Answer from the NTA final answer key. Solution by PrepWiser.

Study Binomial Theorem

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