assume the producer has 5 songs and 5 commercials available. in how many ways can the producer create a morning show?

To determine how many ways the producer can create a morning show using 5 songs and 5 commercials, we first need to define the structure of the morning show. Let's assume:

- The producer selects 3 songs from the 5 available.
- The producer selects 2 commercials from the 5 available.

### Step 1: Number of ways to choose 3 songs from 5 songs
The number of ways to choose 3 songs from 5 can be calculated using the combination formula:

\[
\text{Number of ways to choose 3 songs} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

### Step 2: Number of ways to choose 2 commercials from 5 commercials
Similarly, the number of ways to choose 2 commercials from 5 can be calculated using the same combination formula:

\[
\text{Number of ways to choose 2 commercials} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

### Step 3: Total number of ways to create the morning show
Since the selections of songs and commercials are independent, we multiply the number of ways to choose the songs and the number of ways to choose the commercials:

\[
\text{Total number of ways} = \binom{5}{3} \times \binom{5}{2} = 10 \times 10 = 100
\]

Thus, there are **100 different ways** for the producer to create the morning show.

---

Would you like further clarification or details on any part of the calculation?

### Related questions:
1. What if the producer wants to select all 5 songs and commercials?
2. How would the number of ways change if the producer must include exactly 1 commercial?
3. What if the order of songs and commercials matters? How would you calculate that?
4. How does the calculation change if the producer has 6 songs but still selects only 3?
5. Can you explain the difference between permutations and combinations in this context?

**Tip:** When dealing with combinations, the order of selection does not matter, unlike in permutations where order is important.

Discover how to calculate the total number of ways a producer can create a morning show by selecting 3 songs from 5 available songs and 2 commercials from 5 available commercials. This problem involves combinations and explores the use of the combination formula. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation