📘 Chapter 2.2 – Counting Techniques

This section introduces powerful methods for counting possible outcomes in experiments, especially when listing them is not practical. Understanding these techniques is essential for calculating probabilities.

🔢 1. Multiplication Rule

If an operation consists of a sequence of k steps, and:

Then the total number of ways the operation can be done is:

n₁ × n₂ × ... × nₖ

Example: A website has 4 color options, 3 font styles, and 3 image positions.
Total Designs = 4 × 3 × 3 = 36

📐 2. Permutations

Permutations count the number of ways to arrange items in order. For n different items, the number of permutations is:

n! = n × (n - 1) × (n - 2) × ... × 1

📌 Permutations of r elements from n (order matters)

P(n, r) = n! / (n - r)!

Example: Place 4 components into 8 positions: P(8, 4) = 8 × 7 × 6 × 5 = 1680

🧮 3. Combinations

Combinations count groups where order does not matter. The number of combinations is:

C(n, r) = n! / (r!(n - r)!)

Example: Choose 5 positions out of 8 (order doesn't matter):
C(8, 5) = 56

🔁 Permutations vs. Combinations — What’s the Difference?

Key idea: Permutations care about order; Combinations do not.

Concept Order? Formula Example (n=3: A,B,C; r=2)
Permutation Order matters P(n, r) = n! / (n - r)! AB, BA, AC, CA, BC, CB → 6
Combination Order does not matter C(n, r) = n! / (r!(n - r)!) {A,B}, {A,C}, {B,C} → 3

Rule of thumb: If swapping positions creates a new outcome, use permutations. If swapping positions gives the same outcome, use combinations.

Try it: Compare permutations vs. combinations for the same n and r.

🧪 4. Sampling Without Replacement

What if we want a sample with specific criteria?

Example: In a bin of 50 parts (3 defective), how many 6-part samples have exactly 2 defectives? 
Step 1: Choose 2 from 3 defectives:
C(3, 2) = 3! / (2! · 1!) 
        = (3 × 2 × 1) / [(2 × 1)(1)]
        = 6 / 2
        = 3

Step 2: Choose 4 from 47 non-defectives:
C(47, 4) = 47! / (4! · 43!) 
         = (47 × 46 × 45 × 44) / (4 × 3 × 2 × 1)
         = 4,280,760 / 24
         = 178,365

Total samples (multiply independent choices):
= C(3,2) × C(47,4)
= 3 × 178,365
= 535,095