๐Ÿ”ข Counting Principles: Permutations & Combinations

1. Fundamental Counting Principle

The Fundamental Counting Principle states that if one event can occur in m ways and a second event can occur in n ways, then the number of ways both events can occur together is m ร— n.

This principle extends to multiple events: if there are multiple independent choices to make, multiply the number of options for each choice.

Total Ways = nโ‚ ร— nโ‚‚ ร— nโ‚ƒ ร— ... ร— nโ‚–
Example:
A restaurant offers 4 appetizers, 6 main courses, and 3 desserts. How many different three-course meals can you order?

Solution: 4 ร— 6 ร— 3 = 72 different meals

๐Ÿ“ Practice Question

A password consists of 2 letters followed by 3 digits. How many different passwords are possible? (26 letters, 10 digits)
A) 2,600
B) 67,600
C) 676,000
D) 1,000,000

2. Permutations

A permutation is an arrangement of objects in a specific order. The order matters in permutations.

The number of permutations of n distinct objects taken r at a time is denoted as P(n, r) or โ‚™Pแตฃ.

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

Special case: P(n, n) = n! (all objects arranged)
Remember: n! (n factorial) = n ร— (n-1) ร— (n-2) ร— ... ร— 2 ร— 1
Example: 5! = 5 ร— 4 ร— 3 ร— 2 ร— 1 = 120
Example:
In how many ways can 5 students be arranged in a row for a photo?

Solution: P(5, 5) = 5! = 120 ways

Example:
How many 3-letter "words" can be formed from the letters A, B, C, D, E (without repetition)?

Solution: P(5, 3) = 5!/(5-3)! = 5!/2! = 120/2 = 60 words

๐Ÿ“ Practice Question

A club has 8 members. In how many ways can they elect a president, vice president, and secretary (one person cannot hold multiple positions)?

3. Combinations

A combination is a selection of objects where order does NOT matter. Combinations are used when we're choosing or selecting items.

The number of combinations of n objects taken r at a time is denoted as C(n, r), โ‚™Cแตฃ, or "n choose r" (โฟCแตฃ).

C(n, r) = n! / (r! ร— (n - r)!)
Permutation vs Combination:
โ€ข Use permutations when order matters (arranging, ranking, ordering)
โ€ข Use combinations when order doesn't matter (selecting, choosing, forming a group)
Example:
From a group of 10 people, how many ways can we select a committee of 3?

Solution: C(10, 3) = 10!/(3! ร— 7!) = (10 ร— 9 ร— 8)/(3 ร— 2 ร— 1) = 120 ways

Example:
How many different 5-card hands can be dealt from a standard deck of 52 cards?

Solution: C(52, 5) = 52!/(5! ร— 47!) = 2,598,960 hands

๐Ÿ“ Practice Question

A pizza place offers 10 different toppings. How many different 3-topping pizzas can be made?
A) 30
B) 90
C) 120
D) 720

๐Ÿ“š Quick Reference

Fundamental Counting Principle: Multiply the number of choices
Permutations P(n, r): Order matters โ†’ n!/(n-r)!
Combinations C(n, r): Order doesn't matter โ†’ n!/(r!(n-r)!)