What will be the sign of the product if we multiply together `199` negative and `10` positive integers?

Options

A. `Positive`
B. `Negative`
C. `"both positive and negative"`
D. `"none of these"`

Answer

To determine the sign of the product when multiplying together 199 negative integers and 10 positive integers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Count of Negative and Positive Integers**:
- We have 199 negative integers.
- We have 10 positive integers.

2. **Determine the Sign of the Product of Negative Integers**:
- When multiplying negative integers, the sign of the product depends on whether the count of negative integers is odd or even.
- Since 199 is an odd number, the product of 199 negative integers will be negative.

3. **Determine the Sign of the Product with Positive Integers**:
- Now, we need to consider the multiplication of the negative product with the positive integers.
- The product of any negative number with a positive number is negative.

4. **Combine the Results**:
- Since the product of the 199 negative integers is negative and we are multiplying this by 10 positive integers, the overall product will remain negative.

### Final Conclusion:
The sign of the product when multiplying together 199 negative integers and 10 positive integers will be **negative**.

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