How Many 5-Ball Combinations Can Be Selected from 12 Identical or Unique Red Balls?

When selecting 5 balls from 12 identical red balls, the problem simplifies considerably. All the balls are the same, so the selection can be made in only one way, as the order and distinctiveness of the balls do not matter.

Identical Balls

In the case of identical balls, the selection can be abstracted. Regardless of how you choose them, all selections will look the same because the balls are indistinguishable.

For example, consider the following: How many 5-balls can you choose from a set of 12 identical red balls?

The answer is: There is only 1 way.

Motivated by the idea that any group of five identical balls selected is the same as any other, regardless of how you try to vary the selection process. This principle holds when the balls are indistinguishable and identical.

Unique Balls

However, when the balls are not identical but distinct (e.g., they may have different purchase times or rates of wear), a more complex calculation is required to determine the number of possible combinations.

For instance, if you're selecting 5 balls from 12 unique red balls:

12 x 11 x 10 x 9 x 8  95,040 ways

Here, during your first pick, you have 12 choices. For the second pick, you have 11 remaining choices, for the third pick, 10, for the fourth, 9, and for the fifth, 8.

However, this count includes duplicates, as the order in which you pick the balls does not matter. For instance, selecting ball 1, then ball 2, and so on, is considered the same as selecting ball 2 first, then ball 1, and so on.

To account for this, we need to eliminate repetitions. Each unique group of five balls can be ordered in 5! (5 factorial) ways:

5! 5 x 4 x 3 x 2 x 1 120

Since every group of five balls is uniquely ordered in 120 ways, we divide the total number of selections by 120:

95,040 / 120  792

Hence, there are 792 unique ways to select 5 balls from 12 unique red balls, regardless of the order of selection.

Conclusion

The key difference between identical and unique balls lies in the need to eliminate duplicate orders. When the balls are identical, the number of combinations is simply 1. When the balls are unique, the number of combinations is calculated by considering the total number of permutations and then dividing by the number of permutations for a set of five balls.

This concept can be applied to various real-world scenarios, from lottery draws to complex decision-making processes in various fields such as statistics, probability, and combinatorics.

Understanding these concepts can help in optimizing processes, analyzing data, and making informed decisions. The mathematical principles discussed here form a solid foundation for such applications.