Question

The coefficient of $x^6$ in the expansion of $(1+3x-2x^3)^6$ is

• A.
  • 831
• B.
  • 2401
• C.
  • 2451
• D. 2401

Answer 1

Answer: (C)

• 2451

Answer 2

The general term in the expansion of $(1+3x-2x^3)^6$ is given by the multinomial theorem: $\frac{6!}{n_1!n_2!n_3!} (1)^{n_1} (3x)^{n_2} (-2x^3)^{n_3}$ where $n_1+n_2+n_3=6$. We need the coefficient of $x^6$, so $n_2+3n_3=6$.

Possible non-negative integer solutions for $(n_1, n_2, n_3)$ are:

1. $n_3=0 \implies n_2=6$. Then $n_1=0$. Term: $\frac{6!}{0!6!0!} 3^6 (-2)^0 = 729$.
2. $n_3=1 \implies n_2=3$. Then $n_1=2$. Term: $\frac{6!}{2!3!1!} 3^3 (-2)^1 = 60 \cdot 27 \cdot (-2) = -3240$.
3. $n_3=2 \implies n_2=0$. Then $n_1=4$. Term: $\frac{6!}{4!0!2!} 3^0 (-2)^2 = 15 \cdot 1 \cdot 4 = 60$.

The total coefficient is the sum: $729 - 3240 + 60 = -2451$.