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

Options

A. `500/1001`
B. `503/1001`
C. `505/1001`
D. `101/1001`

Answer

To solve the problem of finding the probability that the product of four randomly chosen numbers (from 6 positive and 8 negative numbers) is positive, we can follow these steps:

### Step 1: Calculate the Total Number of Ways to Choose 4 Numbers
The total number of numbers available is \(6 + 8 = 14\). We need to choose 4 numbers from these 14.

The total number of ways to choose 4 numbers from 14 is given by the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

\[ \text{Total ways} = \binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14!}{4! \cdot 10!} \]

Calculating this gives:

\[ \binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = \frac{24024}{24} = 1001 \]

### Step 2: Determine Favorable Outcomes for a Positive Product
To have a positive product, we can consider the following cases:

#### Case 1: All 4 Numbers are Positive
We can choose all 4 from the 6 positive numbers:

\[ \text{Ways} = \binom{6}{4} \cdot \binom{8}{0} = \binom{6}{4} \cdot 1 = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

#### Case 2: All 4 Numbers are Negative
We can choose all 4 from the 8 negative numbers:

\[ \text{Ways} = \binom{6}{0} \cdot \binom{8}{4} = 1 \cdot \binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

#### Case 3: 2 Positive and 2 Negative Numbers
We can choose 2 from the 6 positive and 2 from the 8 negative numbers:

\[ \text{Ways} = \binom{6}{2} \cdot \binom{8}{2} = \frac{6!}{2! \cdot 4!} \cdot \frac{8!}{2! \cdot 6!} = 15 \cdot 28 = 420 \]

### Step 3: Total Favorable Outcomes
Now, we sum the favorable outcomes from all cases:

\[ \text{Total Favorable Outcomes} = 15 + 70 + 420 = 505 \]

### Step 4: Calculate the Probability
The probability that the product is positive is given by the ratio of favorable outcomes to total outcomes:

\[ P(\text{positive product}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{505}{1001} \]

### Final Answer
Thus, the probability that the product of the four chosen numbers is positive is:

\[ \frac{505}{1001} \]