The Combination Formula and Why it Works | Combinatorics, Combinations and Permutations

How many ways can we select k objects from a collection of n objects? The number of combinations of k objects from a collection of n objects has a very nice formula that we will demonstrate and explain in today's video combinatorics lesson!

I assume you're mostly familiar with factorials and permutations in this lesson, if you need to brush up on either topic, check out these lessons!

Factorials Lesson:    • What is That Exclamation Point (What ...  

Permutations Lesson:    • The Permutation Formula and Why it Wo...  

The combination formula is also called a binomial coefficient because it appears in the powers of binomials. It's very cool and we'll talk more about that in another lesson!

SOLUTION TO PRACTICE PROBLEM:

How many ways can we select 4 actors from a group of 7? That would be 7 choose 4, which is 7!/((7-4)!*4!) = (7*6*5*4)/4! = 35.

How about 7 choose 3? That is 7!/((7-3)!*3!) = (7*6*5)/3! = 35.

Look at that, they're equal! Why is that? You can think of it like this. The number of ways we can select 4 actors from a group of 7 is the same as the number of ways we can select 3 actors to NOT be chosen for the play. If we choose the actors that won't get roles, we have necessarily chosen the remaining actors to get roles as well. Another example is that 8 choose 2 is equal to 8 choose 6. If we select 2 objects from a group of 8, we have in a way also chosen the other 6 to not be selected. So the number of ways of choosing 2 objects from 8 is the same as the number of ways of choosing 6 from 8! Can you generalize this principle?



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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