Video example: 
The Organic Chemistry Tutor Permutations and Combinations Tutorial
Example
To illustrate the difference between permutations and combinations, let's use the four elements A, B, C, and D, and pick 2 out of 4. This example will clearly show how these two concepts differ.
Permutations
Permutations are arrangements where the order matters. When we pick 2 out of 4 elements and the order is important, we're dealing with permutations.
- n!/(n-r!)
- nPr
Number of ways to order r objects from a collection of n objects
For our elements A, B, C, and D, the possible permutations when picking 2 are:
AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC
Total number of permutations: 12
Combinations
Combinations are selections where the order doesn't matter. When we pick 2 out of 4 elements and we only care about which elements are chosen, not their order, we're dealing with combinations.
- nCr
- “n choose r”, also known as binomial coefficient
- n!/(n-r)! r! = nPr/r!
Number of ways to choose r objects from a collection of n objects
For our elements A, B, C, and D, the possible combinations when picking 2 are:
AB, AC, AD, BC, BD, CD
Total number of combinations: 6
Order, number of outcomes differ between combinations and permutations
- Order: In permutations, AB and BA are considered different. In combinations, AB and BA are considered the same.
- Number of outcomes: There are more permutations than combinations for the same set of elements.
- Formula:
- Permutations: P(4,2) = 4!/(4-2)! = 12
- Combinations: C(4,2) = 4!/(2!(4-2)!) = 6
- Use case: Use permutations when the order is important (like lock combinations), and combinations when only the selection matters (like lottery numbers).
This example clearly shows that when order matters (permutations), we have twice as many possibilities compared to when order doesn't matter (combinations).