Solution: This is a multiset permutation problem with 7 drones: 3 identical multispectral (M), 2 thermal (T), and 2 LiDAR (L). The number of distinct sequences is:
Solution to the Multiset Permutation Problem: Arranging 7 Drones with Repeated Types
In combinatorics, permutations of objects where some items are identical pose an important challenge—especially in real-world scenarios like drone fleet scheduling, delivery routing, or surveillance operations. This article solves a specific multiset permutation problem featuring 7 drones: 3 multispectral (M), 2 thermal (T), and 2 LiDAR (L) units. Understanding how to calculate the number of distinct sequences unlocks deeper insights into planning efficient drone deployment sequences.
Problem Statement
We are tasked with determining the number of distinct ways to arrange a multiset of 7 drones composed of:- 3 identical multispectral drones (M),- 2 identical thermal drones (T),- 2 identical LiDAR drones (L).
We seek the exact formula and step-by-step solution to compute the number of unique permutations.
Understanding Multiset Permutations
When all items in a set are distinct, the number of permutations is simply \( n! \) (factorial of total items). However, when duplicates exist (like identical drones), repeated permutations occur, reducing the count.
The general formula for permutations of a multiset is:
\[\frac{n!}{n_1! \ imes n_2! \ imes \cdots \ imes n_k!}\]
where:- \( n \) is the total number of items,- \( n_1, n_2, \ldots, n_k \) are the counts of each distinct type.
Applying the Formula to Our Problem
From the data:
- Total drones, \( n = 3 + 2 + 2 = 7 \)- Multispectral drones (M): count = 3- Thermal drones (T): count = 2- LiDAR drones (L): count = 2
Plug into the formula:
\[\ ext{Number of distinct sequences} = \frac{7!}{3! \ imes 2! \ imes 2!}\]
Step-by-step Calculation
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Compute \( 7! \):\( 7! = 7 \ imes 6 \ imes 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 5040 \)
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Compute factorials of identical items:\( 3! = 6 \)\( 2! = 2 \) (for both T and L)
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Multiply denominators:\( 3! \ imes 2! \ imes 2! = 6 \ imes 2 \ imes 2 = 24 \)
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Divide total permutations by repeated permutations:\[\frac{5040}{24} = 210\]
Final Answer
The number of distinct sequences to arrange the 7 drones — 3 multispectral (M), 2 thermal (T), and 2 LiDAR (L) — is:
\[\boxed{210}\]
This formula and result apply broadly to scheduling tasks, logistics routing, or any scenario requiring unique permutations of labeled but repeated units.
Why This Matters
Accurately calculating drone deployment sequences helps maximize operational efficiency, avoid scheduling conflicts, and ensure balanced use of different sensor types in field operations. Mastery of multiset permutations enables smarter planning and resource allocation in drone-based systems.
Key Takeaways
- When arranging objects with repeated elements, factor out duplicates using factorials.- Use \( \frac{n!}{k_1! \cdot k_2! \cdot \cdots \cdot k_m!} \) for sets with \( m \) distinct categories.- This approach efficiently solves complex sequencing problems across robotics, logistics, and computer science.
For further optimization and analysis of drone fleet operations, consider extending the model to include dynamic routing or time-dependent assignments—leveraging these combinatorial foundations for precision in real-world deployment.
