Welcome to permutations and combinations! These are fundamental counting principles in mathematics. Permutations count arrangements where order matters, while combinations count selections where order doesn't matter. Let's explore these concepts using a simple example with three cards labeled A, B, and C.
Permutations count arrangements where order matters. The formula is P of n comma k equals n factorial divided by n minus k factorial. For example, if we want to arrange 2 cards from our 3 cards A, B, and C, we get P of 3 comma 2 equals 6. Here are all 6 possible arrangements: A-B, A-C, B-A, B-C, C-A, and C-B. Notice that A-B is different from B-A because order matters in permutations.
Combinations count selections where order doesn't matter. The formula is C of n comma k equals n factorial divided by k factorial times n minus k factorial. For example, if we want to choose 2 cards from our 3 cards, we get C of 3 comma 2 equals 3. Here are the 3 possible combinations: A and B, A and C, and B and C. Notice that we don't count A-B and B-A as different because order doesn't matter in combinations.
The key difference between permutations and combinations is whether order matters. For permutations, order matters, so we get more arrangements. For combinations, order doesn't matter, so we get fewer selections. With 3 items choosing 2, permutations give us 6 arrangements: A-B, A-C, B-A, B-C, C-A, and C-B. But combinations give us only 3 selections: A-B, A-C, and B-C, because A-B is the same as B-A when order doesn't matter.
To summarize what we've learned: Permutations count arrangements where order matters, while combinations count selections where order doesn't matter. Use P of n comma k equals n factorial over n minus k factorial for permutations, and C of n comma k equals n factorial over k factorial times n minus k factorial for combinations. The key question to always ask is: does order matter in this problem? This will guide you to choose the correct counting method.