The value of `((50),(6))-((5),(1))((40),(6))+((5)/(2))((30),(6))-((5),(3))((20),(6))+((5),(4))((10),(6))` where `((n),(r ))` denotes `"^(n)C_(r )`, is

Options

A. `15625`
B. `0`
C. `1000000`
D. `2250000`

Explanation

To solve the expression

\[ \binom{50}{6} - \binom{5}{1}\binom{40}{6} + \frac{5}{2}\binom{30}{6} - \binom{5}{3}\binom{20}{6} + \binom{5}{4}\binom{10}{6} \]

we will use the Binomial Theorem and properties of binomial coefficients.

### Step 1: Rewrite the expression using binomial coefficients

The expression consists of binomial coefficients, which we can denote as:

\[ \binom{50}{6} - 5 \cdot \binom{40}{6} + \frac{5}{2} \cdot \binom{30}{6} - 10 \cdot \binom{20}{6} + 5 \cdot \binom{10}{6} \]

### Step 2: Apply the Binomial Theorem

According to the Binomial Theorem, we can express \((1 + x)^n\) as:

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

### Step 3: Calculate the coefficients

We need to find the coefficients of \(x^6\) in the expansions of the following:

1. \( (1 + x)^{50} \)
2. \( (1 + x)^{40} \)
3. \( (1 + x)^{30} \)
4. \( (1 + x)^{20} \)
5. \( (1 + x)^{10} \)

### Step 4: Set up the expression

Using the coefficients from the expansions, we can rewrite the expression as:

\[ \text{Coefficient of } x^6 \text{ in } (1 + x)^{50} - 5 \cdot \text{Coefficient of } x^6 \text{ in } (1 + x)^{40} + \frac{5}{2} \cdot \text{Coefficient of } x^6 \text{ in } (1 + x)^{30} - 10 \cdot \text{Coefficient of } x^6 \text{ in } (1 + x)^{20} + 5 \cdot \text{Coefficient of } x^6 \text{ in } (1 + x)^{10} \]

### Step 5: Calculate the coefficients

Using the formula \(\binom{n}{r}\):

- Coefficient of \(x^6\) in \((1 + x)^{50}\) is \(\binom{50}{6}\)
- Coefficient of \(x^6\) in \((1 + x)^{40}\) is \(\binom{40}{6}\)
- Coefficient of \(x^6\) in \((1 + x)^{30}\) is \(\binom{30}{6}\)
- Coefficient of \(x^6\) in \((1 + x)^{20}\) is \(\binom{20}{6}\)
- Coefficient of \(x^6\) in \((1 + x)^{10}\) is \(\binom{10}{6}\)

### Step 6: Substitute back into the expression

Now substituting these coefficients back into the expression gives:

\[ \binom{50}{6} - 5 \cdot \binom{40}{6} + \frac{5}{2} \cdot \binom{30}{6} - 10 \cdot \binom{20}{6} + 5 \cdot \binom{10}{6} \]

### Step 7: Calculate the final value

Now we can calculate each binomial coefficient:

1. \(\binom{50}{6} = 15,890,700\)
2. \(\binom{40}{6} = 3,838,380\)
3. \(\binom{30}{6} = 593,775\)
4. \(\binom{20}{6} = 38,760\)
5. \(\binom{10}{6} = 210\)

Substituting these values into the expression:

\[ 15,890,700 - 5 \cdot 3,838,380 + \frac{5}{2} \cdot 593,775 - 10 \cdot 38,760 + 5 \cdot 210 \]

Calculating each term:

- \(15,890,700\)
- \(- 19,191,900\)
- \( \frac{5}{2} \cdot 593,775 = 1,484,437.5\)
- \(- 387,600\)
- \(1,050\)

Now summing these values:

\[ 15,890,700 - 19,191,900 + 1,484,437.5 - 387,600 + 1,050 = 2254 \]

### Final Answer

The value of the expression is:

\[ \boxed{2254} \]