We want to purchase a set of music CDs and we have a choice of 2 different types of contemporary music CDs and 1 classical music CD. Therefore, we have to consider two events.
A: choice of 2 types of contemporary music CDs
B: choice of 1 classical music CD
First, let's find how many choices of 2 types of contemporary music CDs we have. We want to choose 2 types out of 5 possible. Notice the order in which we choose the types of music is not important, since we are interested only in different types of them. Let's recall the formula for the number of combinations of n objects taken r at a time, where r ≤ n.
_nC_r = n!/(n-r)! * r!In our case we can choose from is 5 different types, so n = 5. Out of them we choose 2 CDs, therefore r = 2. Let's substitute these values into the formula.
We found that there are 10 different combinations of types of contemporary music CDs we can choose. Similarly, let's find the number of combinations of 1 classical music CD out of 3 possible. Therefore, we need to evaluate _3C_1.
We found that there are 3 different choices of one classical music CD. Notice the choice of contemporary music is independent from the choice of the classical music. That enables us to use the Fundamental Counting Principle, since this principle is used to find the number of possible outcomes for a combination of independent events.
Fundamental Counting Principle
If an event A has n possible outcomes and an event B has m possible outcomes, then the total number of different outcomes for A and B combined is n * m.
Therefore, to find how many different sets of music types we can choose for our purchase, we need to multiply the number of combinations of choices of contemporary music CDs and one classical CD.
Let's substitute the obtained values and evaluate the number of possible different sets of music types we can choose for our purchase.
10 * 3 = 30
