Fundamental Counting & Permutations
Ever wondered how many possible license plates exist? The Fundamental Counting Principle gives us the answer! When you have multiple decisions to make, multiply the number of ways to make each decision.
For example, if an ice cream shop offers 3 cone sizes, 15 flavors, and 8 toppings, you can create 3 × 15 × 8 = 360 different cones. Similarly, Virginia license plates with three letters followed by four digits have 26³ × 10⁴ = 17,576,000 possibilities!
Factorials are another important counting tool, representing the product of all natural numbers from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Remember that 0! equals 1, which is a special case.
Permutations help us count arrangements where order matters. The formula is ⁿPᵣ = n!/!, where n is the total objects available and r is how many you're using. When arranging all objects , you can simply use n!. For instance, arranging 7 students in first, second, and third place gives ⁷P₃ = 7!/! = 210 possibilities.
Quick Tip: Think about whether order matters in your problem. For arranging students in a line or determining finishing positions in a race, order definitely matters—that's a permutation!
