Single correct · +4 / −1

(2x - 3y)⁴ = ... The coefficient of x³y is:

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

Answer (b) C(4,3) × 2³ × (-3)¹

Solution

Maths sir · 4 steps

The power of y tells you r, and the exponents of 2 and -3 must add up to 4.

  1. General term. Treat 2x as the first term and -3y as the second. The (r+1)th term of (2x - 3y)⁴ is below.

    Tᵣ₊₁ = C(4, r)(2x)⁴⁻ʳ(-3y)ʳ

  2. Pick r. For x³y the power of y is 1, so r = 1 and the power of 2x is 4 - 1 = 3. The coefficient is C(4, 1) × 2³ × (-3)¹.
  3. Symmetry. Since C(n, r) = C(n, n-r), you have C(4, 1) = C(4, 3) = 4. So the coefficient is 4 × 8 × (-3) = -96.
  4. So. Option (b), C(4,3) × 2³ × (-3)¹, writes this coefficient in its full general-term form and equals -96.

The idea to keep. Carry the number and its sign along with the variable: (2x) and (-3y), not just x and y. The exponents of the two terms always add up to n.

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

Study Binomial Theorem

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