The coefficient of `x^53` in `sum_(r = 0)^(100) ""^100C_r (x - 3)^(100 - r) .2^r` is

Options

A. `""^100C_47`
B. `""^100C_53`
C. `""^(-100)C_53`
D. `""^100C_100`

Answer

To find the coefficient of \( x^{53} \) in the expression

\[ \sum_{r=0}^{100} \binom{100}{r} (x - 3)^{100 - r} \cdot 2^r, \]

we can simplify the expression using the Binomial Theorem.

### Step-by-Step Solution:

1. **Recognize the Binomial Expansion**: The given sum can be rewritten as the expansion of \( (x - 3 + 2)^{100} \) using the Binomial Theorem. This is because:
\[ (x - 3 + 2)^{100} = (x - 1)^{100}. \]

2. **Apply the Binomial Theorem**: According to the Binomial Theorem, we can expand \( (x - 1)^{100} \):
\[ (x - 1)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} (-1)^k. \]

3. **Identify the Required Coefficient**: We need to find the coefficient of \( x^{53} \) in this expansion. The term corresponding to \( x^{53} \) will occur when \( 100 - k = 53 \), which implies:
\[ k = 100 - 53 = 47. \]

4. **Calculate the Coefficient**: The coefficient of \( x^{53} \) is given by:
\[ \binom{100}{47} (-1)^{47}. \]

5. **Simplify the Expression**: Since \( (-1)^{47} = -1 \), the coefficient simplifies to:
\[ -\binom{100}{47}. \]

Thus, the coefficient of \( x^{53} \) in the given expression is

\[ -\binom{100}{47}. \]

### Final Answer:
The coefficient of \( x^{53} \) is \( -\binom{100}{47} \).

---