Combinations

When the order doesn't matter, it is a Combination.

When the order does matter it is a Permutation.

"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

Combinations without repetitions

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!

No Repetition: such as lottery numbers (2,14,15,27,30,33)

Number of combinations

where n is the number of things to choose from, and we choose r of them, no repetition, order doesn't matter.

It is often called "n choose r" (such as "16 choose 3"). And is also known as the Binomial Coefficient.

Combinations with repetitions

Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)

Or let us say there are five flavours of ice cream: banana, chocolate, lemon, strawberry and vanilla.

We can have three scoops. How many variations will there be?

Let's use letters for the flavours: {b, c, l, s, v}. Example selections include:

{c, c, c} (3 scoops of chocolate)
{b, l, v} (one each of banana, lemon and vanilla)
{b, v, v} (one of banana, two of vanilla)

Number of combinations

Where n is the number of things to choose from, and we choose r of them. Repetition allowed, order doesn't matter.

Cheat Sheets

Permutations cheat sheet

Combinations cheat sheet

Permutations/combinations algorithm ideas.

References

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