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Combinatorics is a branch of mathematics dealing with the study of countable, discrete structures. It helps us count the number of ways certain events can occur. In the context of our CD problem, combinatorics allows us to calculate the total number of possible ways to choose and arrange songs.

Using combinatorics, we can determine the total number of combinations of playing 4 songs out of 13, which is essential to find the probabilities we are interested in. This number is given by the binomial coefficient, which is a key concept we will discuss next.

To get a better understanding of combinatorics, think about how many ways you can rearrange a set of items. For example:

• If you have two songs, you can arrange them in 2 different ways.
• If you have three songs, the number of arrangements increases further.

Combinatorics simplifies complex counting problems like these, enabling us to solve probability questions more efficiently.

The binomial coefficient is a mathematical term used to describe the number of ways to choose a given number of elements from a larger set. It is denoted as \(\binom{n}{k}\), and it plays a crucial role in calculating probabilities in combinatorics.

In our CD problem, we need to figure out how many ways we can choose 4 songs out of the 13 available. This is calculated using the binomial coefficient:

\[ \binom{13}{4} = \frac{13!}{4!(13-4)!} = 715 \]

Here, the exclamation mark (\(!)\) denotes a factorial, which means multiplying a series of descending natural numbers. For example, 5 factorial (\(5!\)) equals 5 × 4 × 3 × 2 × 1.

Using binomial coefficients, we can also find out how many ways to select liked and unliked songs among the first 4 played. For instance:

• The number of ways to choose 2 liked songs from 5 liked ones is \(\binom{5}{2}\)
• The number of ways to choose 2 unliked songs from 8 unliked ones is \(\binom{8}{2}\)

Multiplying these results gives us the total number of favorable combinations, which we then use to find the probability.

Favorable outcomes refer to the specific outcomes that satisfy the conditions of a probability problem. In our CD example, favorable outcomes are the combinations of songs that match the number of liked songs we want to play among the first 4.

For instance, to find the probability of liking exactly 2 out of the first 4 songs:

• First, calculate the number of ways to choose 2 liked songs from 5 liked ones: \(\binom{5}{2} = 10\)
• Next, calculate the number of ways to choose 2 unliked songs from 8 unliked ones: \(\binom{8}{2} = 28\)

So, the number of favorable combinations is \(10 \times 28 = 280\).

To get the probability, we divide the number of favorable outcomes by the total number of outcomes. In this case:

\[ P(\text{like 2 out of 4}) = \frac{280}{715} = \frac{4}{11} \]

This concept of finding favorable outcomes is repeated for different conditions, like liking 3 out of 4 songs or liking all 4 songs. Each time, the process involves calculating how many combinations fulfill our criteria and then getting the probability by dividing by the total possible combinations.