https://file.aiquickdraw.com/mathbot/file/1729383971194wnhnse0i.jpg

The problem in the image asks the following:

**How many permutations of three items can be selected from a group of six?**

You are also asked to:
- Use the letters **A, B, C, D, E, and F** to identify the items.
- List all permutations of the specific items **B, D, and F**.

### Step 1: Find the number of permutations
The formula for finding the number of permutations of \( r \) items from a set of \( n \) items is given by:

\[
P(n, r) = \frac{n!}{(n - r)!}
\]

In this case:
- \( n = 6 \) (since you have 6 items: A, B, C, D, E, F)
- \( r = 3 \) (you are choosing 3 items)

So, the number of permutations is:

\[
P(6, 3) = \frac{6!}{(6 - 3)!} = \frac{6 \times 5 \times 4 \times 3!}{3!} = 6 \times 5 \times 4 = 120
\]

Thus, there are **120 permutations** of three items from a group of six.

### Step 2: List the permutations of B, D, and F
The number of permutations of the specific items **B, D, F** is:

\[
3! = 3 \times 2 \times 1 = 6
\]

These six permutations are:
1. BDF
2. BFD
3. DBF
4. DFB
5. FBD
6. FDB

### Summary:
- There are **120 total permutations** of any three items selected from the six.
- The six permutations of **B, D, and F** are: BDF, BFD, DBF, DFB, FBD, and FDB.

Do you have any questions or would you like more details?

### Related Questions:
1. How many combinations are there when selecting 3 items from 6 without considering the order?
2. What is the difference between permutations and combinations?
3. How would the solution change if you were asked to permute 4 items from a group of 6?
4. Can you calculate the total number of permutations for 5 items from a set of 7?
5. What are the permutations if the selected letters were **A, B, and C** instead of **B, D, and F**?

### Tip:
For permutation problems, always keep in mind that the **order matters**—this distinguishes them from combinations.

How many permutations of three items can be selected from a group of six? Use the letters A, B, C, D, E, and F to identify the items, and list each of the permutations of items B, D, and F.

Explore how to calculate the number of permutations when selecting three items from a group of six, using the letters A, B, C, D, E, and F. This problem involves key combinatorial concepts and is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation