`((100 - 1) (100 - 2) (100 - 3)...(100 - 200))/(100 xx 99 xx 98 xx...xx 3 xx 2 xx 1)` is equal to

Options

A. `(100)/(99 xx 98 xx 97 xx... xx 3 xx 2 xx 1)`
B. `- (1)/(99 xx 98 xx 97 xx ... xx 3 xx 2 xx 1)`
C. 0
D. `-(2)/(99 xx 98 xx 97 xx...xx 3 xx 2 xx 1)`

Answer

To solve the expression \(\frac{(100 - 1)(100 - 2)(100 - 3) \ldots (100 - 200)}{100 \times 99 \times 98 \ldots \times 3 \times 2 \times 1}\), let's break it down step by step.

### Step 1: Understand the Numerator
The numerator is the product of terms from \(100 - 1\) to \(100 - 200\). This can be expressed as:
\[ (100 - 1)(100 - 2)(100 - 3) \ldots (100 - 200) \]
This means we have:
\[ 99 \times 98 \times 97 \times \ldots \times (-100) \]
Notice that \(100 - 100 = 0\). Therefore, one of the terms in the product is \(0\).

### Step 2: Identify the Impact of Zero
Since the product includes a term that is \(0\), the entire product becomes \(0\):
\[ (100 - 1)(100 - 2)(100 - 3) \ldots (100 - 200) = 0 \]

### Step 3: Understand the Denominator
The denominator is the factorial of \(100\):
\[ 100 \times 99 \times 98 \ldots \times 3 \times 2 \times 1 = 100! \]
However, since the numerator is \(0\), the denominator does not affect the overall value of the expression.

### Step 4: Final Calculation
Now, substituting back into the expression, we have:
\[ \frac{0}{100!} = 0 \]

### Conclusion
Thus, the value of the entire expression is:
\[ \boxed{0} \]