Question

The coefficient of \( x^3 \) in the expansion of \[ \frac{1}{(1 + 2x)^{-10}} \] is:

• A. 980
• B. 960
• C. 1020
• D. 860
• E. 880

Hint:

When dealing with binomial expansions of the form \( (1 + ax)^n \), use the binomial theorem and identify the term corresponding to the power of \( x \) you are interested in. Remember that \( \binom{n}{k} a^k x^k \) represents the general term.

Answer 1

The Correct Option is B

We are asked to find the coefficient of \( x^3 \) in the expansion of \( \frac{1}{(1 + 2x)^{-10}} \). 
We can write the expression as: \[ \frac{1}{(1 + 2x)^{-10}} = (1 + 2x)^{10} \] Now, apply the binomial expansion to \( (1 + 2x)^{10} \). 
The binomial expansion for \( (1 + 2x)^n \) is given by: \[ (1 + 2x)^{10} = \sum_{k=0}^{10} \binom{10}{k} (2x)^k \] 
We need to find the coefficient of \( x^3 \). 
This corresponds to the term where \( k = 3 \) in the expansion: \[ \binom{10}{3} (2x)^3 \] \[ = \binom{10}{3} \cdot 2^3 \cdot x^3 \] 
Now, calculate \( \binom{10}{3} \) and \( 2^3 \): \[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \] \[ 2^3 = 8 \] Thus, the coefficient of \( x^3 \) is: \[ 120 \times 8 = 960 \] Thus, the correct answer is option (B), 960.