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

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:
  • $5! = 120$
• Restricted Arrangements (Alice and Ben together):
  • Treat Alice and Ben as one unit: $4!$ arrangements
  • Alice and Ben can switch places: $2!$
  • Total restricted arrangements: $24 \times 2 = 48$
• Final Calculation:
  • Subtract restricted from total: $120 - 48 = 72$

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'

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: $\frac{11!}{2! \times 2! \times 2!}$
  • Calculation: $\frac{39916800}{8} = 4989600$
• Arrangements with M's Adjacent:
  • Treat 'MM' as one unit, reducing to "MMATHEATICS".
  • Formula: $\left( \frac{10!}{2! \times 2!} \right) \times 2!$
  • Calculation: $\left( \frac{3628800}{4} \right) \times 2 = 1814400$
• Final Calculation:
  • Subtract arrangements with 'M's adjacent from total.
  • Calculation: $4989600 - 1814400 = 3175200$

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: $3! = 6$
• Restrictions (A and B in Same Row):
  • $2 \times 2! = 4$ (for A and B together in one row)
• Final Calculation:
  • Total without restriction: $6 \times 6 = 36$
  • Restricted (A and B together): $4 \times 2 = 8$
  • Total with restriction: $36 - 8 = 28$

Result: $28$ distinct arrangements.