# Of 19, possible books, you take 3 with you on vacation. How many different collections of 3 books can you take?

Of 19

possible​ books, you plan to take 3

with you on vacation. How many different collections of 3

books can you​ take?

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• It depends on your suitcase weight allowance?

• I won't argue Zac's calculation; but a mathematical question is not a question about books. It's about math. You could ask that question regarding three CDs, three rings, three suits, and Zac's answer would be the same. It does not relate to jewelry, clothing or compact disks any more than booking a hotel suite related to books.

So why don't you direct your question to the Math category?

• 19 x 18 x 17 : 3! = 5,814 : 6 = 969

The first part is easy to understand: For the first book you have 19 choices. For each of these 19 different choices you have another 18 choices (a total of 19 x 18 = 342), and for each of these two-book combinations you have then a further 17 choices for the third book.

That gets you to the 5,814 three book collections.

However (!), you will have some duplicate collections because they will differ only in the order the books were chosen. E.g., if you choose books 4, 13, and 6, this will effectively be the same collection than if you choose books 13, 4, and 6, or books 4, 6, and 13, etc.

The number of different arrangements, called permutations, is calculated by taking the factorial of the number of elements. Here, it's 3! ("three factorial") equaling 6.

So unless the order of books is important, in which case you have 5,814 different, ordered collections, you'll have to divide the result by 6 in order to eliminate the collections which differ only by their book order.

That gives you 969 different sets of books.

Another way to arrive at this result is to calculate the combination directly using the binomial coefficient, in this case "19 choose 3".

The formula looks slightly different but is effectively the same:

"19 choose 3"* = 19! : [(19-3)! x 3!]

* I can't use the notation in this format, basically a 19 placed over a 3 in two big round brackets.

• Unless it was a long vacation, I'd be lucky to get thru one.

1.  History - one on the US Civil War

2.  Murder mystery

3.  Fiction