Solution: We compute the number of distinct permutations of a multiset with 13 total batches: 6 blue (B), 4 green (G), and 3 red (R). The number of sequences is: - RTA
Understanding the Number of Distinct Permutations of a Multiset: A Case Study with 6 Blue, 4 Green, and 3 Red Batches
Understanding the Number of Distinct Permutations of a Multiset: A Case Study with 6 Blue, 4 Green, and 3 Red Batches
When dealing with sequences composed of repeated elements, calculating the number of distinct permutations becomes essential in fields like combinatorics, data science, and algorithm optimization. A classic example is determining how many unique sequences can be formed using multiset batches鈥攕uch as 6 blue, 4 green, and 3 red batches鈥攖otaling 13 batches.
The Problem: Counting Distinct Permutations of a Multiset
Understanding the Context
Given a multiset with repeated items, the total number of distinct permutations is computed using the multinomial coefficient. For our case:
- Blue (B): 6 units
- Green (G): 4 units
- Red (R): 3 units
- Total: 6 + 4 + 3 = 13 batches
The formula to compute the number of distinct permutations is:
\[
\ ext{Number of permutations} = \frac{13!}{6! \cdot 4! \cdot 3!}
\]
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Key Insights
Where:
- \(13!\) is the factorial of the total number of batches, representing all possible arrangements if all elements were unique.
- The denominators \(6!, 4!, 3!\) correct for indistinguishable permutations within each color group鈥攖he overcounting that occurs when swapping identical elements.
Why Use the Multinomial Coefficient?
Without accounting for repetitions, computing permutations of 13 objects would yield \(13! = 6,227,020,800\) arrangements鈥攂ut this overcounts because swapping the 6 identical blue batches produces no new distinct sequence. Dividing by \(6!\), \(4!\), and \(3!\) removes the redundant orderings within each group, giving the true number of unique sequences.
Applying the Formula
Now compute step-by-step:
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\[
\frac{13!}{6! \cdot 4! \cdot 3!} = \frac{6,227,020,800}{720 \cdot 24 \cdot 6}
\]
Calculate denominator:
\(720 \ imes 24 = 17,280\), then \(17,280 \ imes 6 = 103,680\)
Now divide:
\(6,227,020,800 \div 103,680 = 60,060\)
Final Result
The number of distinct permutations of 6 blue, 4 green, and 3 red batches is:
60,060 unique sequences
Practical Applications
This calculation supports a wide range of real-world applications, including:
- Generating all possible test batch combinations in quality control
- Enumerating permutations in random sampling designs
- Optimizing scheduling and routing when tasks repeat
- Analyzing DNA sequencing data with repeated nucleotides
Conclusion
When working with multiset permutations, the multinomial coefficient provides a precise and efficient way to count distinct arrangements. For 13 batches with multiplicities of 6, 4, and 3, the total number of unique sequences is 60,060鈥攁 clear example of how combinatorial math underpins problem-solving across science and engineering disciplines.
