Problem:
There are 6 positive and 8 negative numbers. Four numbers are selected at random without replacement and multiplied. Find the probability that the product is positive.

Solution:
To find the probability that the product of the four randomly selected numbers is positive, we need to consider two cases:
1. The product is positive when all four numbers are positive.
2. The product is positive when three numbers are negative and one number is positive.

Case 1: All four numbers are positive
In this case, we need to select all four positive numbers. There are 6 positive numbers to choose from, and we need to select 4 of them. The number of ways to select 4 positive numbers from 6 is given by the combination formula:
C(6,4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15

Case 2: Three numbers are negative and one number is positive
In this case, we need to select 3 negative numbers and 1 positive number. There are 8 negative numbers to choose from, and we need to select 3 of them. The number of ways to select 3 negative numbers from 8 is given by the combination formula:
C(8,3) = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56

We also need to select 1 positive number from the remaining 6 positive numbers, which can be done in C(6,1) = 6 ways.

Total number of favorable outcomes:
The total number of favorable outcomes is the sum of the number of favorable outcomes from each case:
Total favorable outcomes = 15 + (56 * 6) = 15 + 336 = 351

Total number of possible outcomes:
To find the total number of possible outcomes, we need to select 4 numbers from the total 14 numbers (6 positive + 8 negative). The number of ways to select 4 numbers from 14 is given by the combination formula:
C(14,4) = 14! / (4!(14-4)!) = 14! / (4!10!) = (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1) = 1001

Probability:
The probability that the product of the four randomly selected numbers is positive is given by:
Probability = Total favorable outcomes / Total possible outcomes = 351 / 1001 = 0.35065 (approximately)

Conclusion:
The probability that the product of the four randomly selected numbers is positive is approximately 0.35065.