Question

Coefficient of $ x^{11}$ in the expansion of $(1 + x^2)^4 (1 + x^3)^7 (1 + x^4)^{12} $

• A. 1051
• B. 1106
• C. 1113
• D. 1120

Answer 1

The Correct Option is C

The correct answer is C:1113
The expression can be rewritten as a sum of terms involving various coefficients:
\([ (1+x^2)^4 \cdot (1+x^3)^7 \cdot (1+x^4)^{12} ]\)
\( = (4C_0 + 4C_1 x^2 + 4C_2 x^4 + 4C_3 x^6 + 4C_4 x^8) \cdot (7C_0 + 7C_1 x^3 + 7C_2 x^6 + 7C_3 x^9 + \ldots + 7C_7 x^{21}) ]\)
\([ \cdot (12C_0 + 12C_1 x^4 + 12C_2 x^8 + \ldots + 12C_{12} x^{48}) ]\)
The objective is to find the term that contains \(( x^{11} )\) which corresponds to a power of ( x ) equal to 11. This can be achieved by selecting appropriate combinations of coefficients from each part\(= (4C_0 \cdot 7C_1 \cdot 12C_2) + (4C_1 \cdot 7C_3 \cdot 12C_0) + (4C_2 \cdot 7C_1 \cdot 12C_1) + (4C_4 \cdot 7C_1 \cdot 1) ]\)
Calculating these products of coefficients:
\( = [(1 \cdot 7 \cdot 66) + (4 \cdot 35 \cdot 1) + (6 \cdot 7 \cdot 12) + (1 \cdot 7) = 462 + 140 + 504 + 7 = 1113 ]\)
So, the coefficient of \(( x^{11} )\) in the given expression is 1113.
In a more natural language:
The expression is composed of three parts, each raised to a certain power, and it's required to determine the coefficient of \(( x^{11} )\) in the expanded
form. This involves selecting appropriate coefficients from each part. After calculating the necessary products of coefficients, the coefficient of \(( x^{11} )\)is found to be 1113.