Question

If \( (1 + x + x^2)^{10} = 1 + a_1 x + a_2 x^2 + \dots \), then \( (a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 \) equals to:

• A. 0
• B. 10
• C. 20
• D. 30

Hint:

In binomial expansions, group the terms based on the power of \( x \) and use symmetry to simplify the calculation of coefficients.

Answer 1

The Correct Option is C

We are given the expansion: \[ (1 + x + x^2)^{10} = 1 + a_1 x + a_2 x^2 + a_3 x^3 + \dots \] We need to find the value of \( (a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 \). The general term in the expansion of \( (1 + x + x^2)^{10} \) is given by: \[ T_k = \binom{10}{k} x^k + \binom{10}{k+1} x^{k+1} + \binom{10}{k+2} x^{k+2} \] To find the coefficients \( a_n \) for different powers of \( x \), we use the binomial expansion formula and identify the relevant terms. When the sum is \( (a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 \), we can simplify by recognizing that the coefficients alternate based on the nature of the powers of \( x \) (odd/even). After simplifying and calculating the terms, we get: The required value is \( 20 \). Thus, the answer is \( 20 \).