Question

I have three distinct mystery novels, three distinct fantasy novels, and three distinct biographies. I'm going on vacation, and I want to take two books of different genres. How many possible pairs can I choose?

Answer 1

`((3),(2))((3),(1))^2=3xx3^2=27`

Explanation:

Let's first notice that this is a combination problem because we don't care in what order we pick the books (if we did, that'd be a permutation problem).

The general formula for a combination is:

`C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

To do this, let's first see that we're choosing 2 categories from the population of 3 categories. That's

`((3),(2))`

For each of the categories, there are 3 books we can choose from and we want 1. For each category chosen, then, we have:

`((3),(1))`

Since we have 2 categories chosen, we need to square this:

`((3),(1))^2`

And so altogether we have:

`((3),(2))((3),(1))^2=3xx3^2=27`

We can list them out (I'll use A, B, and C; L, M, and N; and W, X, and Y to list out the books):

AL BL CL
AM BM CM
AN BN CN

AW BW CW
AX BX CX
AY BY CY

LW MW NW
LX MX NX
LY MY NY