Question

If the ratio of the terms equidistant from the middle term in the expansion of \( (1 + x)^{12} \) is \( \frac{1}{256} \), then the sum of all the terms of the expansion \( (1 + x)^{12} \) is:

• A. \( 4^{12} \) or \( 6^{12} \)
• B. \( 3^{12} \) or \( 5^{12} \)
• C. \( 6^{12} \) or \( 7^{12} \)
• D. \( 12^{12} \)

Hint:

For binomial expansions, use the symmetry of the terms and the given ratio to relate the terms equidistant from the middle term to solve for the unknowns.

Answer 1

The Correct Option is B

Step 1: Binomial Expansion. The expansion of \( (1+x)^{12} \) is given by: \[ (1+x)^{12} = \sum_{k=0}^{12} \binom{12}{k} x^k. \] Step 2: Equidistant Terms. The ratio of the equidistant terms from the middle term is given as \( \frac{1}{256} \). 
From this, we deduce that the sum of all terms is \( 512 \).