The study of permutations and combinations in mathematics offers a multitude of intriguing problems, particularly when it comes to arranging objects with certain restrictions.
Restricted Arrangements
- Objective: To arrange items with specific rules or limitations.
- Application: Useful in complex problems, especially in combinatorics.
Key Concepts
1. Fixed Position: Some items must be in specific places.
2. Adjacency: Certain items must or must not be next to each other.
3. Separation: Specific spacing between items is required.
Strategy for Solving
- Total Minus Restricted: First calculate all possible arrangements, then subtract those that break the rules.
Examples
Example 1. Arranging People in a Line
![People in line](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F5d6b102f-2921-4087-95d7-1b24f64731d4-file.png&w=1080&q=75)
Image courtesy of Vertor Stock
Objective: Arrange Alice, Ben, Carol, Dave, and Emma in a line with the restriction that Alice and Ben are not next to each other.
Solution:
- Total Unrestricted Arrangements:
- Restricted Arrangements (Alice and Ben together):
- Treat Alice and Ben as one unit: arrangements
- Alice and Ben can switch places:
- Total restricted arrangements:
- Final Calculation:
- Subtract restricted from total:
Result: There are 72 ways to arrange these five people with Alice and Ben not next to each other.
Example 2. Arranging Letters in 'MATHEMATICS'
![Mathematics](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fe595994e-1bf1-41c0-8702-7effb6d92e37-file.png&w=1080&q=75)
Image courtesy of Brainly
Objective: Arrange the letters in "MATHEMATICS" such that the two 'M's are never adjacent.
Solution:
- Total Arrangements Without Restriction:
- Formula:
- Calculation:
- Arrangements with M's Adjacent:
- Treat 'MM' as one unit, reducing to "MMATHEATICS".
- Formula:
- Calculation:
- Final Calculation:
- Subtract arrangements with 'M's adjacent from total.
- Calculation:
Result: There are 3,175,200 distinct arrangements of "MATHEMATICS" where the two 'M's are not adjacent.
Example 3: Multi-row Arrangements
Objective: Arrange six students in two rows of three, with restrictions on placements.
Solution:
- Each Row Arrangements:
- Restrictions (A and B in Same Row):
- (for A and B together in one row)
- Final Calculation:
- Total without restriction:
- Restricted (A and B together):
- Total with restriction:
Result: distinct arrangements.