What are combinations?
Conceptually, the combinations task is easy to understand: it’s about the number of ways to select a subgroup from a whole group. As in the permutations task, the whole group is notated as n, and the subgroup is notated as r. The standard formula for combinations adds just one piece to the standard formula for permutations: the expression r! in the denominator:
The combinations standard formula:
Combinations standard formula: n!/r!(n-r)!
Considering that r is the number of elements in the selected subgroup, and going back to our understanding of ordering, r! is the number of ways to order the elements in the selected subgroup. Since the permutations task is “selecting and ordering” and combinations task is “selecting without ordering,” it makes sense that we get from permutations to combinations by dividing out the number of ways to order the things we selected.
Thus, the combinations standard formula counts the number of different subgroups of r things that can be selected from a whole group of n things. The permutations standard formula counts the number of different orders of r things selected from a whole group of n things. The expression r! is the number of orders per subgroup. If we divide the number of orders (given by the permutations formula) by the number of orders per subgroup (r!), we obtain the number of subgroups (the combinations formula).
number of orders/number of orders per subgroup = number or subgroups
Let’s return one last time to our example with the cars:
How many ways can 4 cars be selected from a collection of 7 cars?
n = 7 and r = 4, so we can plug these values into the standard formula:
n!/r!(n-r)!
7!/4!(7-4)! = 35
In effect, we just performed the permutations task (counted the number of orders for four cars selected from seven cars) and divided out the number of orders per subgroup (4!) to obtain an answer of 35 possible subgroups of fourcars.
Let’s put this connection between combinations and permutations in real terms. When we selected and ordered four cars from the group of seven (performing the permutations task), we might have done it like this:
Aston Martin Bentley Corvette DeLorean
But we also might have done it like this:
Corvette Aston Martin DeLorean Bentle
Same four cars, different order. With our understanding of ordering, we know that there are 4! ways to line up these same four cars. Since the combinations task is about identifying unique subgroups rather than unique orders, we must divide out the number of ways to line up each different subgroup of r things: r!
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Contributor: Elijah Mize (Apex Instructor)