Definition
When you pick a few items out of many, it's a permutation if the order matters. It's a combination if you just form a group and the order doesn't matter. Even with the exact same ingredients, the number of possible outcomes changes completely depending on whether you line them up in a row or toss them into one bowl.
Lock Codes vs. Pizza Toppings
If your bike lock code is 1-2-3, entering 3-2-1 won't open it. A different order of the same digits makes a completely different code. This way of counting, where you care not only about what you picked but also the order you put them in, is called a permutation.
Now picture ordering a pizza with olives, mushrooms, and onions. Whether you tell the cook about the mushrooms first or the olives first, you get the same pizza. When only which kinds you chose matters, no matter the order, you use a combination.
In other words, even if you pick the same number of items from the same group, the moment order becomes a condition, permutations produce far more possible outcomes.
Why Do the Numbers Change When You Calculate?
Say you want to choose a class president and a vice president from 5 students. Any of the 5 can be president. Then the vice president comes from the remaining 4, so the number of outcomes is 5 times 4, which is 20. The president and vice president have set roles and an order, so this is a permutation.
Now let's pick 2 of the 5 students for cleanup duty. Picking Amy and then Ben, or Ben and then Amy, gives the same cleanup pair. Inside those 20 permutations, every pair shows up twice, once in each order, like twins.
That's why, to find a combination, you divide by the number of ways to reorder the chosen items. Divide 20 by 2, and you get 10 ways to pick the cleanup crew.
A Bit More Precisely: Everyday Life and Math Notation
To write ordering calculations in a shorter way, math uses the exclamation mark (!) as the symbol for the factorial. The number of ways to line up all 3 people in a row is 3 times 2 times 1, which makes 6. We write that as "3!".
A combination is a permutation divided by this lining-up count (the factorial). For example, a lottery where you match 6 numbers out of 45 doesn't care about the order you wrote the numbers down, so it's a classic example of a combination. If the lottery were a permutation, where you had to match the exact order the numbers were drawn, your chance of winning first prize would be 720 times smaller than it is now.
In the end, there is one simple question that tells permutations and combinations apart: "If the order changes, does the result change?" If the answer is yes, order matters, so use a permutation. If not, use a combination.
Common misconceptions
The "combination" in "combination lock" means exactly the same thing as a combination in math.
A lock only opens if the numbers are in the right order, so in math it isn't a combination at all. It's a permutation, where order counts. Since you can reuse the same digit, the exact term is a "permutation with repetition." The everyday word "combination" and the math term differ on whether order is considered.
Where you meet it
Giving gold, silver, and bronze medals to 1st, 2nd, and 3rd place in a race is a permutation, while choosing 3 athletes for a relay team is a combination.
A 4-digit phone passcode is a permutation, because order counts (and since digits can repeat, it's exactly a permutation with repetition). Picking 6 lottery numbers is a combination.
In one sentence
If order matters and you line things up, it's a permutation. If you just pick a group and order doesn't matter, it's a combination.