Coefficient of `x^(48)` in `sum_(r=0)^(50)""^(50)C_(r).(x-2)^(x).3^(50-r)` is

Options

A. `.^(50)C_(2)`
B. `.^(48)C_(2)`
C. `3^(48)`
D. none of these

Answer

To find the coefficient of \( x^{48} \) in the expression

\[ \sum_{r=0}^{50} \binom{50}{r} (x-2)^r \cdot 3^{50-r}, \]

we can follow these steps:

### Step 1: Rewrite the Expression
We can rewrite the expression using the binomial theorem. The given expression can be interpreted as the expansion of \( (3 + (x-2))^{50} \).

\[ (3 + (x-2))^{50} = (x + 1)^{50}. \]

### Step 2: Expand Using the Binomial Theorem
Using the binomial theorem, we expand \( (x + 1)^{50} \):

\[ (x + 1)^{50} = \sum_{k=0}^{50} \binom{50}{k} x^k \cdot 1^{50-k}. \]

### Step 3: Identify the Coefficient of \( x^{48} \)
From the expansion, we need to find the coefficient of \( x^{48} \). This coefficient corresponds to \( k = 48 \):

\[ \text{Coefficient of } x^{48} = \binom{50}{48}. \]

### Step 4: Simplify the Coefficient
Using the property of binomial coefficients, we know that:

\[ \binom{50}{48} = \binom{50}{2}. \]

### Final Answer
Thus, the coefficient of \( x^{48} \) in the original expression is:

\[ \binom{50}{2}. \]