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.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.