Problem Setup

We need to find the probability of picking exactly 4 green balls and 2 blue balls when selecting 6 balls randomly from a box.

• Total green balls: 5
• Total blue balls: 10
• Total balls: 5 + 10 = 15
• Number of balls to pick: 6
• Desired outcome: 4 green balls and 2 blue balls

Total Possible Outcomes

The total number of ways to choose 6 balls from the 15 available balls is calculated using combinations:

Total combinations = $ \binom{15}{6} $

$ \binom{15}{6} = \frac{15!}{6!(15-6)!} = \frac{15!}{6!9!} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 5005 $

So, there are 5005 possible ways to pick 6 balls from the box.

Favorable Outcomes

The number of ways to pick exactly 4 green balls from the 5 available green balls is:

$ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5 $

The number of ways to pick exactly 2 blue balls from the 10 available blue balls is:

$ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 $

To find the total number of favorable outcomes (picking 4 green AND 2 blue balls), we multiply these two values:

Favorable outcomes = $ \binom{5}{4} \times \binom{10}{2} = 5 \times 45 = 225 $

Probability Calculation

The probability is the ratio of favorable outcomes to total possible outcomes:

Probability = $ \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} $

Probability = $ \frac{225}{5005} $

Simplify the Probability

We simplify the fraction by dividing the numerator and the denominator by their greatest common divisor. Both are divisible by 5:

Probability = $ \frac{225 \div 5}{5005 \div 5} = \frac{45}{1001} $

Conclusion

The probability of picking 4 green balls and 2 blue balls is $ \frac{45}{1001} $. This corresponds to Option B.