Question

The coefficient of \( x^{12} \) in the expansion of \( \left(1+2 x^{2}-x^{3}\right)^{8} \) is 

(1) 6340 

(2) 6342 

(3) 6344 

(4) 6346

Answer 1

\((1 +2x^2 - x^3)^8 = \, ^8C_0 + \,^8C_1(2x^2 - x^3) + \,^8C_2(2x^2 - x^3)^2\\ + \,^8C_3(2x^2 - x^3)^3 + \,^8C_4(2x^2 - x^3)^4 + \,^8C_5(2x^2 - x^3)^5\\+ \,^8C_6(2x^2 - x^3)^6 +\,^8C_7(2x^2 - x^3)^7 + \,^8C_8(2x^2 - x^3)^8\)

Coeffiecient of x¹² = \(^8C_4.\,^4C_4 2^0.(-1)^4 + \,^8C_5.\,^5C_2.2^3.(-1)^2 + \,^8C_6 .^6C_0.2^6(-1)^0\)

\(= \frac{8.7.6.5}{4.3.2.1} + \frac{8.7.6}{3.2.1} \times 8 + \frac{8.7}{2.1}\times64\)

\(= 70 + 448 + 1792\)

\(= 2310\)