Question

The coefficient of \( x^{24} \) in the expansion of \( (1+x^{2})^{12}(1+x^{12})(1+x^{24}) \) is

• A. $^{12}C_{6}$
• B. $^{12}C_{6}+2$
• C. $^{12}C_{6}+4$
• D. $^{12}C_{6}+6$

Hint:

Multiply corresponding powers of $x$ and add their coefficients.

Answer 1

The Correct Option is B

Step 1: Expansion
\( (1+x^2)^{12} = \sum_{r=0}^{12} {}^{12}C_r (x^2)^r \). The expression is \( [1 + {}^{12}C_1 x^2 + \cdots + {}^{12}C_6 x^{12} + \cdots + {}^{12}C_{12} x^{24}] \times (1 + x^{12} + x^{24} + x^{36}) \).
Step 2: Identifying \( x^{24} \) Terms
Possible combinations to get \( x^{24} \): - \( 1 \times x^{24} \) coefficient = 1 - \( {}^{12}C_6 x^{12} \times x^{12} \) coefficient = \( {}^{12}C_6 \) - \( {}^{12}C_{12} x^{24} \times 1 \) coefficient = \( {}^{12}C_{12} = 1 \)
Step 3: Total Coefficient
Total = \( {}^{12}C_6 + 1 + 1 = {}^{12}C_6 + 2 \).
Final Answer: (b)