Permutations and Combinations

Ever wondered how many different outfits you can create from your wardrobe? That's where counting theory comes in! When events can occur in multiple ways, we multiply those possibilities together.

For example, with 3 pants, 5 shirts, 3 shoes, 2 hats, and 4 scarves, you'd have 3×5×3×2×4 = 360 different outfit combinations! Similarly, a Subway restaurant with various options for bread, meats, cheese, and other ingredients can create over 53 million different meal combinations.

The key difference between permutations and combinations is whether order matters. Permutations (P) count arrangements where order is important (like phone numbers), while combinations (C) count selections where order doesn't matter (like lottery numbers). The formulas are:

• Permutations: P = n!/$n-r$!
• Combinations: C = n!/$r!(n-r)!$

Pro Tip: Think about whether order matters in your problem. If rearranging the same items creates a different outcome, you need permutations. If not, use combinations.

Let's see these concepts in action: If 10 people are eligible for 3 prizes, there are C₁₀³ = 120 different ways to distribute them (combination). But if 6 people need to arrange themselves in a 6-seat car, there are 6! = 720 different seating arrangements (permutation). For a swim team selecting 5 girls from 10 and 7 boys from 15, there are (10C5)(15C7) = 1,621,620 possible team combinations!