Combinations

Lessons

Combinations: Learn

A combination is an arrangement in which order does not matter. The notation for combinations is C(n,r) which is the number of combinations of "n" things if only "r" are selected.

For example, in most card games, the order of the cards in your hand does not matter, only which cards are in your hand.

Since order does not matter, that means there are fewer possibile combinations compared to permutations. We can actually divide the permutations by r! to find the number of combinations. The final formula for combinations is:

C(n,r) =
n!
r!(n-r)!
= number of total possible combinations

For example, there is a card game where you must make the best 5-card hand out of 7 available cards.

C(7,5) =
7!
5!(7-5)!
=
7*6*5*4*3*2*1
5*4*3*2*1*2*1
=
7*6*5*4*3*2*1
5*4*3*2*1*2*1
=
7*6
2*1
=
42
2
= 21 possibilities.

Notice how after writing out the factorial in the fraction, you can start to reduce the fraction by canceling most of the factors. This makes factorials the easy way to find how many possible permutations are available.

Combination: Practice

Find the number of combinations.

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C(,) =

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How fast can you get 20 more correct answers than wrong answers? 999

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