Unit 5: Permutations and Combinations

1. Fundamental Principle of Counting
2. Factorial Notation
3. Permutations (Arrangement)
4. Combinations (Selection)
5. Key Differences and Applications
6. Exam Focus Enhancements

1. Fundamental Principle of Counting

This principle forms the basis of all counting logic. It states that if one event can occur in m ways and a second independent event can occur in n ways, then:

Multiplication Principle: The total number of ways the two events can occur together is m × n.
Addition Principle: The number of ways either of the two events can occur is m + n (provided events are mutually exclusive).

2. Factorial Notation

The product of the first n natural numbers is called n factorial, denoted by n!.

n! = n × (n-1) × (n-2) \dots × 3 × 2 × 1
Note: 0! = 1 and 1! = 1.

3. Permutations (Arrangement)

A Permutation is an arrangement in a specific order of a number of objects taken some or all at a time. Order matters in permutations.

ⁿP_r = (n!) / ((n-r)!)
(Where 0 ≤ r ≤ n)

Arrangement of n distinct objects: n!
Permutations when objects are not all distinct: (n!) / (p!q!r!) where p, q, r are counts of identical objects.

4. Combinations (Selection)

A Combination is a selection of a number of objects where the order does not matter. We are only interested in which objects are chosen, not their sequence.

ⁿC_r = (n!) / (r!(n-r)!)
(Where 0 ≤ r ≤ n)

Key Properties:

ⁿC_r = ⁿC_n-r (Symmetry property)
ⁿC₀ = ⁿC_n = 1
Relationship: ⁿP_r = ⁿC_r × r!

5. Key Differences and Applications

Choosing the right formula depends on the problem context:

Keywords

Feature	Permutation	Combination
Focus	Arrangement / Sequence	Selection / Grouping
Order	Matters (AB ≠ BA)	Does not matter (AB = BA)	Arrange, Order, Rank, Digits	Select, Choose, Group, Committee

6. Exam Focus Enhancements

Exam Tips

Logic Check: If the problem asks for a Committee or a Team, use Combinations (ⁿC_r). If it asks for Passwords or Phone numbers, use Permutations (ⁿP_r).
Calculation Hack: To calculate ⁿC_r, multiply the first r numbers of n downwards and divide by r!.
Example: ¹⁰C₃ = (10 × 9 × 8) / (3 × 2 × 1).
Circular Permutations: If n objects are arranged in a circle, the number of ways is (n-1)!.

Common Mistakes

Repetition: Watch out for "repetition allowed" vs "repetition not allowed." Standard ⁿP_r assumes no repetition. If allowed, it's n^r.
0! Value: Many students think 0! = 0. Remember, it is always **1**.
Double Counting: In selection problems, don't use permutations or you will overcount identical groups in different orders.

Frequently Asked Questions

Q: When do I use addition instead of multiplication?
A: Use multiplication when events must happen one after another (And). Use addition when you have different "cases" or options (Or).

Q: What is Pascal's Rule?
A: ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r. This is frequently used in proof-based questions.