Find coeff. of `x^(25)` in the expansion of
`sum_(k=0)^(50)(-1)^(k)""^(50)C_(k)(2x-3)^(50-k)(2-x)^(k)`.

Answer

To find the coefficient of \( x^{25} \) in the expansion of

\[ \sum_{k=0}^{50} (-1)^k \binom{50}{k} (2x - 3)^{50-k} (2 - x)^k, \]

we can follow these steps:

### Step 1: Recognize the Binomial Expansion
The given expression can be recognized as a binomial expansion. We can rewrite the summation in a more manageable form. Notice that it resembles the binomial theorem, where:

\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k. \]

In our case, we can set \( x = 2x - 3 \) and \( y = 2 - x \).

### Step 2: Combine Terms
We can combine the terms inside the summation:

\[ (2x - 3) + (2 - x) = 2x - 3 + 2 - x = x - 1. \]

Thus, we can rewrite the original expression as:

\[ \sum_{k=0}^{50} (-1)^k \binom{50}{k} (2x - 3)^{50-k} (2 - x)^k = (x - 1)^{50}. \]

### Step 3: Expand the Expression
Now we need to find the coefficient of \( x^{25} \) in the expansion of \( (x - 1)^{50} \). According to the binomial theorem:

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

### Step 4: Identify the Required Coefficient
We are interested in the coefficient of \( x^{25} \). From the binomial expansion, the coefficient of \( x^{25} \) is given by:

\[ \binom{50}{25} (-1)^{50-25} = \binom{50}{25} (-1)^{25}. \]

### Step 5: Simplify the Coefficient
Since \( (-1)^{25} = -1 \), we have:

\[ \text{Coefficient of } x^{25} = -\binom{50}{25}. \]

### Final Answer
Thus, the coefficient of \( x^{25} \) in the expansion is:

\[ -\binom{50}{25}. \]