Why do we divide Permutations to get to Combinations?

Solution 1:

Maybe, looking at an example clarifies this best :

You have $20$ objects and have to choose $5$ of them. How many possibilities are there ?

You have $20,19,18,17,16$ choices explaining how $\frac{20!}{15!}$ comes into the play.

Now each combination can appear in $5!$ possible orders which correspond to the same combination. Therefore we have to divide by $5!$ to find the number of combinations.

This can be generalized to arbitary numbers explaining how the binomial coefficient emerges.

Solution 2:

Let's bunch together permutations that have the same content. Each resulting equivalence class has $k!$ members. Therefore, the number of such classes is $k!$ times smaller than the number of permutations. A combination is just an equivalence class or, if you prefer, one nominated element thereof.