11.1 Permutations vs Combinations

📖 Main Content

🔑 The Core Difference

Understanding Permutations

A permutation is a way of arranging items where the order matters.

Think of it this way: if you arrange the letters A, B, C, then ABC and BAC are counted as two different arrangements because the order is different.

Real-life example: Imagine a race with 3 runners — Alice, Bob, and Charlie. The results 1st: Alice, 2nd: Bob, 3rd: Charlie is a completely different outcome from 1st: Bob, 2nd: Alice, 3rd: Charlie. The order in which they finish matters. This is a permutation situation.

Other examples where order matters:

• Creating a 4-digit PIN code (1234 ≠ 4321)
• Arranging books on a shelf
• Assigning 1st, 2nd, and 3rd place in a competition
• Forming a password using letters and numbers

The permutation formula:

When you choose r items from a group of n items and arrange them (order matters), the number of ways is:

$^nP_r = \frac{n!}{(n-r)!}$

where n! (called n factorial) means: $n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$

Quick example: How many ways can you arrange 3 letters chosen from A, B, C, D, E?

Here, $n = 5$ and $r = 3$.

$^5P_3 = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = \frac{120}{2} = 60$

So there are 60 different arrangements.