To solve the problem, let's break it down step-by-step.

Step 1: Total Number of Employees

The company has 600 employees in total.

Step 2: Number of Ways to Select 4 Employees

We need to select 4 employees from the 600. The number of ways to choose 4 people out of 600 is given by the combination formula:

$\binom{600}{4} = \frac{600!}{4!(600 - 4)!}$

This represents the total number of possible selections.

Step 3: Number of Ways to Select 4 Employees Including John and Jill

If John and Jill are both selected, we need to choose 2 additional employees from the remaining 598 employees (since John and Jill are already chosen). The number of ways to do this is:

$\binom{598}{2} = \frac{598!}{2!(598 - 2)!}$

Step 4: Probability Calculation

The probability that both John and Jill are selected is the ratio of the favorable outcomes to the total outcomes. Thus, the probability is:

$P(\text{both John and Jill win}) = \frac{\binom{598}{2}}{\binom{600}{4}}$

Step 5: Compute the Values

Now let's compute the combinations.

1. $\binom{600}{4} = \frac{600 \times 599 \times 598 \times 597}{4 \times 3 \times 2 \times 1}$
2. $\binom{598}{2} = \frac{598 \times 597}{2 \times 1}$

We can plug these into the formula to find the exact probability.

Let me calculate these values for you. math

Total number of employees

total_employees = 600

Number of ways to choose 4 employees from 600

total_ways = math.comb(total_employees, 4)

Number of ways to choose 2 employees from the remaining 598 after selecting John and Jill

favorable_ways = math.comb(total_employees - 2, 2)

Probability calculation

probability = favorable_ways / total_ways round(probability, 6)The probability that both John and Jill will win the trip is approximately $0.000033$ (rounded to six decimal places).

Would you like further details or have any questions about this solution?

Here are 5 related questions you may find interesting:

1. How does the combination formula work when selecting groups from a larger population?
2. What is the difference between combinations and permutations?
3. What is the probability of a different couple (other than John and Jill) both winning the trip?
4. How would the probability change if the company selected 6 employees instead of 4?
5. How can we calculate the probability of selecting at least one couple for the trip?

Tip: When dealing with probabilities and combinations, remember that combinations count unordered selections, whereas permutations count ordered selections.