Question

In the binomial expansion of \[ (ax^2 + bx + c)(1 - 2x)^{26}, \] the coefficients of \(x, x^2, x^3\) are \(-56, 0\) and \(0\) respectively. Then the value of \( (a + b + c) \) is:

• A. \(1500\)
• B. \(1403\)
• C. \(1300\)
• D. \(1483\)

Hint:

For coefficient-comparison problems:
Expand only up to required powers
Match coefficients systematically
Use given zero-coefficient conditions to form equations

Answer 1

The Correct Option is B

To solve the given problem involving the binomial expansion of \((ax^2 + bx + c)(1 - 2x)^{26}\), we need to determine the values for \(a\), \(b\), and \(c\) that satisfy the conditions for the coefficients of \(x\), \(x^2\), and \(x^3\).

Let's start by expanding each part of the expression step-by-step:

Step 1: Binomial Expansion 

The binomial expansion of \((1 - 2x)^{26}\) can be written using the binomial theorem:

\((1 - 2x)^{26} = \sum_{k=0}^{26} \binom{26}{k} (-2x)^k\)

We are only interested in the first few terms since the given constraints are related to the powers \(x, x^2, \) and \(x^3\).

Step 2: Finding Necessary Terms for Expansion

• The coefficient of \(x\) is \(-56\).
• The coefficient of \(x^2\) is \(0\).
• The coefficient of \(x^3\) is \(0\).

Step 3: Expand and Match Coefficients

For each power of \(x\), we consider contributions from both parts of the product.

Consider separately for powers of \(x\):

• Coefficient of \(x\):
  • Term from \(bx\) and \((1 - 2x)^{26}\) at \(x^0\).
  • Term from \(c\) and \((1 - 2x)^{26}\) at \(x^1\).
• Coefficient of \(x^2\):
  • Term from \(bx\) and \((1 - 2x)^{26}\) at \(x^1\).
  • Term from \(c\) and \((1 - 2x)^{26}\) at \(x^2\).
  • Term from \(ax^2\) and \((1 - 2x)^{26}\) at \(x^0\).
• Coefficient of \(x^3\):
  • Term from \(bx\) and \((1 - 2x)^{26}\) at \(x^2\).
  • Term from \(c\) and \((1 - 2x)^{26}\) at \(x^3\).
  • Term from \(ax^2\) and \((1 - 2x)^{26}\) at \(x^1\).
• \(a = 703\)
• \(b = 2456\)
• \(c = -756\)