The Core Difference: Order
The fundamental difference between permutations and combinations lies in whether the order of selection matters. In permutations, the order is important, meaning different arrangements of the same items are counted as distinct outcomes. In combinations, the order is irrelevant, and only the final group of items matters, regardless of how they were arranged.
Section 2: Understanding Permutations
A permutation is an arrangement of objects in a specific sequence. It is used when you are counting the number of possible ways to order a set of items. Think of it as lining things up or assigning specific roles. The formula for permutations without repetition is nPr = n! / (n-r)!, where 'n' is the total number of items, and 'r' is the number of items to arrange.
Section 3: Understanding Combinations
A combination is a selection of items from a larger set where the order of selection does not matter. It is used when you are counting the number of possible groups that can be formed. Think of it as choosing a team or a committee. The formula for combinations is nCr = n! / (r!(n-r)!), where 'n' is the total number of items, and 'r' is the number of items to choose.
Section 4: A Practical Example
Imagine you have three friends: Alice (A), Bob (B), and Charlie (C). If you need to award a gold and silver medal (a permutation, as order matters), the possibilities are AB, BA, AC, CA, BC, CB (6 options). If you simply need to choose two friends to join a committee (a combination, as order doesn't matter), the possibilities are just AB, AC, BC (3 options), because the committee 'Alice and Bob' is the same as 'Bob and Alice'.