Question

Find the number of ways to purchase 5 different kinds of drinks from a selection of 13 drinks.



1716


1287


715


336

Answer 1

This is a combination problem and because it is asking the number of ways of doing something. So you want to be able to count how many ways 13 things can be arranged in 5 groups of 5.

While this seems difficult at first, there is a formula that you can use:
$$
13 \choose 5
$$

But if you don't "see" how this counts the number of ways of arranging 13 things in groups of 5 then go to the basics.

You can arrange 13 things in a line in 13! ways.

When you count the number of ways of arranging 13 things in groups of 5, you also need to account for the ways of arranging the remaining 8 things.

Also, there are some groups that are redundant, like abcef and efcba, which have the same "things" in them but just arranged differently. This should just count once, so divide 13! by 5! because for every arrangement of 5 there are 5! ways to arrange them. The takes out the redundant arrangements.

Of the remaining 8 things that aren't part of the arrangement, we need to count them only once so for the same reason we divide by 8!.

That is how we get
$$
\cfrac{13!}{5!8!}=\cfrac{13!}{5!(13-5)!} ={13 \choose 5}
$$

http://www.wolframalpha.com/input/?i=13+choose+5

Does that make sense?

Answer 2

Sort of. How do I get the final answer though?

Answer 3

Never mind I got it thank you so much!!