Question

From a set containing four positive numbers and four negative numbers, four numbers are chosen at random and they are multiplied. The probability that the obtained product is positive is:

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{19}{35} \)
• D. \( \frac{23}{35} \)

Hint:

Remember that a positive product requires an even number of negative factors. Breaking it down systematically into distinct cases (0, 2, or 4 negative numbers) ensures you don't miss any valid combinations.

Answer 1

The Correct Option is C

Step 1: Find the total ways.
Choosing 4 numbers from 8 gives $\binom{8}{4} = 70$ total ways.
Step 2: Know when the product is positive.
A product is positive when the count of negative numbers picked is even: 0, 2, or 4.
Step 3: Case of 0 negatives.
Pick 4 positives: $\binom{4}{0}\binom{4}{4} = 1\times1 = 1$.
Step 4: Case of 2 negatives.
Pick 2 negatives and 2 positives: $\binom{4}{2}\binom{4}{2} = 6\times6 = 36$.
Step 5: Case of 4 negatives.
Pick 4 negatives: $\binom{4}{4}\binom{4}{0} = 1\times1 = 1$. So favorable $= 1+36+1 = 38$.
Step 6: Compute the probability.
\[ P = \frac{38}{70} = \frac{19}{35} \] \[ \boxed{\tfrac{19}{35}} \]