Choosing a class president and vice president out of 5 friends, versus choosing 2 cleaning-duty helpers. Both are "choosing 2 out of 5," so it can feel odd at first that the answer comes out different. For president and vice president, who ends up as president and who as vice president — the order (the role) — matters. But for cleaning duty, all that matters is that two people get picked, so order doesn't matter. This exact difference is what permutations and combinations deal with.
A count where order matters is called a "permutation," written nPr. Choosing 2 out of 5 people and also deciding their order, any of the 5 can fill the first spot, and any of the remaining 4 can fill the second, giving 5×4=20 ways. A count where order doesn't matter is called a "combination," written nCr — and to get it, you take the 20 permutations found above and merge together the cases that are "the same two people, just in a different order." There are 2!=2 ways to arrange two people, so dividing 20 by 2 gives 10, the combination answer.
Generalize this relationship and you get the formula nCr = nPr ÷ r!. A combination is always less than or equal to the matching permutation, and if you remember the reason is exactly "several cases collapse into one because order isn't distinguished," you can derive the formula yourself without ever memorizing it. Interestingly, if you line up the nCr values in order, they match exactly with the numbers in Pascal's Triangle. The r-th number in the n-th row of Pascal's Triangle is exactly nCr. The fact that two concepts starting from completely different places end up meeting at the same numbers is one of the charms of combinatorics.
Permutations and combinations are hiding all over everyday life. A case like a lock's passcode, where a different order gives a completely different result, is a permutation; a case like the lottery, where only which 6 numbers got drawn matters and the order they were drawn in doesn't, is a combination. The reason winning a 6/45 lottery has such a low probability is exactly this: the number of combinations for choosing 6 out of 45 numbers with no regard to order (45C6) is a staggering figure over 8 million. In card games, cases where the exact order of a hand matters, versus cases where you just look at which combination of cards you're holding, are also good real-world examples of permutations and combinations.
When working with kids, it's most effective to have them list things out by hand starting with small numbers. For example, listing out permutations for choosing 2 out of 3 people (A, B, C) gives AB, BA, AC, CA, BC, CB — 6 in total — while looking at it as combinations and merging AB(=BA), AC(=CA), and BC(=CB) each into one gives 3. Having them actually write this out by hand helps them physically understand why the formula looks the way it does. Let them feel how counting by hand gets hard even with just slightly bigger numbers, and then teach the formula — the need for the formula lands naturally too.
On our activity page, you can change the total number of people (n) and the number chosen (r) with sliders, and check the count for both permutations and combinations along with the full listing. When the numbers are small, every case gets listed out directly, so you can count with your own eyes and confirm why a combination is always smaller than the matching permutation.