Combinations and Probability: Picking Marbles from a Box

In this article, we will explore the problem of picking marbles from a unique box. Specifically, we'll calculate the number of ways to pick 5 marbles from a box containing 8 red marbles and 7 green marbles, with the condition that at least 3 of the picked marbles are green. This problem will be approached using the principles of combinations and probability.

Understanding the Problem

The total number of marbles in the box is 15 (8 red 7 green). We need to find the number of ways to pick 5 marbles such that at least 3 of them are green. To solve this, we will break down the problem into three cases based on the number of green marbles picked.

Case Analysis

We can divide the problem into the following cases:

3 green marbles and 2 red marbles 4 green marbles and 1 red marble 5 green marbles and 0 red marbles

Case 1: 3 Green Marbles and 2 Red Marbles

The number of ways to choose 3 green marbles from 7 is given by the combination formula:

C(7, 3) 7! / (3! * (7-3)!) 35

The number of ways to choose 2 red marbles from 8 is given by:

C(8, 2) 8! / (2! * (8-2)!) 28

The total number of ways for this case is:

35 * 28 980

Case 2: 4 Green Marbles and 1 Red Marble

The number of ways to choose 4 green marbles from 7 is:

C(7, 4) C(7, 3) 35

The number of ways to choose 1 red marble from 8 is:

C(8, 1) 8

The total number of ways for this case is:

35 * 8 280

Case 3: 5 Green Marbles and 0 Red Marbles

The number of ways to choose 5 green marbles from 7 is:

C(7, 5) C(7, 2) 7! / (2! * (7-2)!) 21

The total number of ways for this case is:

21

Final Calculation

Now, we sum the total number of ways from all three cases:

Total 980 280 21 1281

Therefore, the total number of ways to pick at least 3 green marbles is

1281

Understanding Combinations

Combinations are a fundamental concept in probability and statistics, used to determine the number of ways to choose a subset of items from a larger set. The combination formula is given by:

C(n, r) n! / (r! * (n-r)!)

a

Additional Cases

For further clarity, let's consider two additional cases:

Case 1: 2 Green Marbles and 1 Red Marble Case 4: 1 Green Marble and 2 Red Marbles

Case 1: 2 Green Marbles and 1 Red Marble

The number of ways to choose 2 green marbles from 7 is:

C(7, 2) 7! / (2! * (7-2)!) 21

The number of ways to choose 1 red marble from 8 is:

C(8, 1) 8

The total number of ways for this case is:

21 * 8 168

Case 4: 1 Green Marble and 2 Red Marbles

The number of ways to choose 1 green marble from 7 is:

C(7, 1) 7

The number of ways to choose 2 red marbles from 8 is:

C(8, 2) 8! / (2! * (8-2)!) 28

The total number of ways for this case is:

7 * 28 196

Conclusion

In conclusion, the total number of ways to pick at least 3 green marbles from the box, considering all cases, is 1281. This problem demonstrates the application of combinations and probability in solving real-world problems.