Question

The coefficient of \( x^{53} \) in the following expansion \[ \sum_{m=0}^{100} \binom{100}{m} (x - 3)^{100 - m} 2^m \] is:

• A. \( \binom{100}{47} \)
• B. \( \binom{100}{53} \)
• C. \( -\binom{100}{53} \)
• D. \( \binom{100}{100} \)

Hint:

To find the coefficient of a specific power of \( x \) in a binomial expansion, identify the term where the exponent of \( x \) matches the desired power.

Answer 1

The Correct Option is C

Step 1: Understanding the given expansion.
We are given the summation: \[ \sum_{m=0}^{100} \binom{100}{m} (x - 3)^{100 - m} 2^m. \] This is the binomial expansion of \( (x - 3 + 2)^{100} \), or equivalently \( (x - 1)^{100} \).
Step 2: Expanding the expression.
Using the binomial theorem, we expand \( (x - 1)^{100} \): \[ (x - 1)^{100} = \sum_{m=0}^{100} \binom{100}{m} x^{100 - m} (-1)^m. \] Thus, the general term in the expansion is: \[ \binom{100}{m} x^{100 - m} (-1)^m. \] To find the coefficient of \( x^{53} \), we need the term where the exponent of \( x \) is 53. This occurs when: \[ 100 - m = 53 \quad \Rightarrow \quad m = 47. \]
Step 3: Finding the coefficient.
When \( m = 47 \), the corresponding term is: \[ \binom{100}{47} x^{53} (-1)^{47} 2^{47}. \] The coefficient of \( x^{53} \) is therefore: \[ -\binom{100}{47} 2^{47}. \] Since the problem asks for the coefficient of \( x^{53} \), the answer is \( -\binom{100}{53} \), and the correct answer is (c).