There are 6 positive numbers and 8 negative numbers. Three numbers are chosen from them at random and multiplied. The probability that the product is a negative number is

Options

A. `(11)/(34)`
B. `(17)/(33)`
C. `(16)/(35)`
D. `(11)/(35)`

Answer

To solve the problem, we need to find the probability that the product of three randomly chosen numbers from a set of 6 positive numbers and 8 negative numbers is negative.

### Step-by-Step Solution:

1. **Determine the Total Number of Ways to Choose 3 Numbers:**
The total number of ways to choose 3 numbers from 14 (6 positive + 8 negative) is given by the combination formula:
\[ \text{Total ways} = \binom{14}{3} \]
Calculating this:
\[ \binom{14}{3} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364 \]

**Hint:** Use the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) to calculate the total ways.

2. **Determine the Favorable Outcomes for a Negative Product:**
A product of three numbers will be negative if:
- All three numbers are negative, or
- Two numbers are positive and one number is negative.

**Case 1: All three numbers are negative.**
The number of ways to choose 3 negative numbers from 8:
\[ \text{Ways} = \binom{8}{3} \]
Calculating this:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

**Case 2: Two positive numbers and one negative number.**
The number of ways to choose 2 positive numbers from 6 and 1 negative number from 8:
\[ \text{Ways} = \binom{6}{2} \times \binom{8}{1} \]
Calculating this:
\[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]
\[ \binom{8}{1} = 8 \]
Thus, the total ways for this case:
\[ 15 \times 8 = 120 \]

3. **Calculate Total Favorable Outcomes:**
Now, we add the favorable outcomes from both cases:
\[ \text{Total favorable outcomes} = 56 + 120 = 176 \]

**Hint:** Make sure to consider all possible combinations that lead to a negative product.

4. **Calculate the Probability:**
The probability that the product is negative is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{negative product}) = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{176}{364} \]
Simplifying this fraction:
\[ \frac{176 \div 8}{364 \div 8} = \frac{22}{46} = \frac{11}{23} \]

5. **Final Answer:**
Thus, the probability that the product of the three chosen numbers is negative is:
\[ \frac{11}{23} \]