Calculating the Total Number of Arrangements of Multicolored Marbles

In a practical scenario, determining the number of ways to arrange a set of items can be an intriguing yet complex problem. Consider a bag containing a specific mix of marbles. In this article, we will explore how to calculate the total number of possible arrangements of 4 white, 5 red, and 3 blue marbles drawn one by one and arranged in a row.

Understanding the Problem

The problem involves finding the total number of distinct arrangements of a set of 12 uniquely colored marbles. Given the mixture of colors, some of the marbles are indistinguishable from each other. Therefore, we must account for the indistinguishable nature of the marbles of the same color when calculating the total permutations.

Formula for Permutations of a Multiset

To solve this problem, we will use the formula for permutations of a multiset. For a set of n objects with groups of indistinguishable objects, the formula is given by:

P(n; n1, n2, n3, ...) n! / n1! * n2! * n3! * ...

Applying the Formula to Our Problem

In the given example, we have a total of 12 marbles:

4 white marbles 5 red marbles 3 blue marbles

We can calculate the total number of arrangements using the given formula:

P(12; 4, 5, 3) 12! / (4! * 5! * 3!)

Step-by-step Calculation

Let's break down the calculation step by step:

Calculate the factorial of 12: 12! 479,001,600 Calculate the factorials of 4, 5, and 3: 4! 24 5! 120 3! 6 Substitute these values into the formula: 12! / (4! * 5! * 3!) 479,001,600 / (24 * 120 * 6) Calculate the denominator: (24 * 120) 2,880 (2,880 * 6) 17,280 Finally, calculate the total number of arrangements: 479,001,600 / 17,280 27,720

Therefore, the total number of ways to arrange the marbles is 27,720.

Alternative Method Using Combinations

Another method to solve this problem involves using combinations. We can break down the arrangement into two steps:

Choose 4 spots out of 12 to place the white marbles: 12C4 From the remaining 8 spots, choose 5 spots to place the red marbles: 8C5

The number of ways to do this is given by:

12C4 * 8C5 27720

This method effectively accounts for the indistinguishability of the marbles and calculates the total number of arrangements.

Conclusion

In conclusion, we have illustrated two methods to calculate the total number of arrangements of 4 white, 5 red, and 3 blue marbles. The final answer for the total number of arrangements is 27,720. This problem showcases the importance of permutations and combinations in solving real-world problems involving indistinguishable objects.