Combinations
Combinations are just the full spread of different ways you can arrange the various subsets of one larger set. For a simple example take the set of A={1,2}. You can form four subsets of this set: {}, {1}, {2}, and {1,2}. These subsets are thus the combinations of set A. Ironically, combinations have nothing to do with combination locks. One important fact about combinations is that order doesn’t matter. So in real life, a combination lock should be called a “permutation lock”, because the order does matter when you’re entering numbers into a lock.You always have fewer combinations that permutations, and here’s why:
Take the numbers 1,2,3,4. If you want to know how many ways you can select 3 items where the order doesn’t matter (and the items aren’t allowed to repeat), you can pick:
123
234
134
124
However, if you want permutations (where the order does matter, the same set has 24 different possibilities. Just take the first combination, 1,2,3 and think of the ways you can order it.
123
132
321
312
231
213
There are six ways to order the numbers, which means there are 4 x 6 ways to order the set of four numbers.
You can use the combinations calculator by entering “n” and “r”. At the bottom of the calculator, you’ll also have an explanation of the answer.
Combinations don’t have to involve numbers — sets can also refer to groups:
Jane, Lin, Gina and Sally is a set of 4.
Pluto, Venus, Mars, Earth and Saturn is a set of 5.
Why do we care about combinations in real life? Combinations have hundreds (possibly, thousands) of applications, the most obvious of which is gambling:
Lottery organizations need to know how many ways numbers can be chosen in order to calculate odds
Slot machine manufacturers need to know how many ways the pictures on the wheels can line up, to calculate odds and prize money
A more advanced use of combinations comes into play with banking, finance and even polling.
