Question

The coefficient of x¹⁰ in the expansion of 1 + (1 + x) + (1 + x)² + … + (1 + x)²⁰ is:

• A. \( 19C9 \)
• B. \( 20C10 \)
• C. \( 21C11 \)
• D. \( 22C12 \)

Hint:

When summing coefficients of \( x^k \) from binomial expansions, use the binomial coefficient formula \( \binom{n}{k} \) to find the required terms.

Answer 1

The Correct Option is C

The given expression is: \[ 1 + (1 + x) + (1 + x)^2 + \cdots + (1 + x)^{20} \] The coefficient of \( x^{10} \) in the expansion of \( (1 + x)^n \) is \( \binom{n}{10} \). The sum of terms is the sum of the coefficients of \( x^{10} \) from each expansion. For the term \( (1 + x)^n \), the coefficient of \( x^{10} \) is \( \binom{n}{10} \). Thus, we need to sum the coefficients of \( x^{10} \) from the expansions \( (1 + x)^1, (1 + x)^2, \dots, (1 + x)^{20} \). The coefficient of \( x^{10} \) in \( (1 + x)^n \) is \( \binom{n}{10} \), and we want to find the sum of all such terms from \( n = 10 \) to \( n = 20 \). The sum will be the binomial coefficient \( \binom{21}{11} \). Thus, the correct answer is \( 21C11 \).