Question

In the binomial expansion of (x - 2y²)⁹, the coefficient of x⁶y⁶ is equal to

• A. -672
• B. 672
• C. 336
• D. -336
• E. -512

Answer 1

The Correct Option is A

In the binomial expansion of \( (x - 2y^2)^9 \), we use the binomial theorem: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Here, \( a = x \), \( b = -2y^2 \), and \( n = 9 \). The general term in the expansion is: \[ T_k = \binom{9}{k} x^{9-k} (-2y^2)^k = \binom{9}{k} x^{9-k} (-2)^k (y^2)^k \] Simplifying: \[ T_k = \binom{9}{k} (-2)^k x^{9-k} y^{2k} \] We need the term where the powers of \( x \) and \( y \) are 6 and 6, respectively. Thus, we set: \[ 9 - k = 6 \quad \text{(for } x^6\text{)} \] This gives \( k = 3 \). Now, we calculate the corresponding term: \[ T_3 = \binom{9}{3} (-2)^3 x^{6} y^6 \] The coefficient is: \[ \binom{9}{3} (-2)^3 = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \times (-8) = 84 \times (-8) = -672 \] 

The correct option is (A) : \(-672\)

Answer 2

We are given the binomial expansion of (x - 2y²)⁹. We want to find the coefficient of x⁶y⁶.

The general term in the binomial expansion of (a + b)ⁿ is given by:

T_k+1 = \(\binom{n}{k} a^{n-k} b^k\)

In our case, a = x, b = -2y², and n = 9. We want to find the term with x⁶y⁶. This means we need n - k = 6 and 2k = 6, so k = 3.

Let's find the term T₃₊₁ = T₄:

T₄ = \(\binom{9}{3} x^{9-3} (-2y^2)^3 = \binom{9}{3} x^6 (-2)^3 (y^2)^3\)

T₄ = \(\binom{9}{3} x^6 (-8) y^6\)

Now we need to calculate \(\binom{9}{3}\):

\(\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} = \frac{9 \times 8 \times 7 \times 6!}{3 \times 2 \times 1 \times 6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 3 \times 4 \times 7 = 84\)

So, T₄ = 84 × x⁶ × (-8) × y⁶ = -672x⁶y⁶

Therefore, the coefficient of x⁶y⁶ is -672.

Therefore, the answer is -672.