https://file.aiquickdraw.com/mathbot/file/1727361894820zy2gankm.jpg

The solution in the image explains a combinatorial problem where four persons enter a bus with 6 vacant seats, and we need to find the number of ways they can take their places.

### Steps to solve:

1. **First Person's Choice**:
   - The first person can choose any of the 6 vacant seats. So, there are **6 choices** for the first person.

2. **Second Person's Choice**:
   - After the first person takes a seat, only 5 vacant seats remain. Therefore, the second person has **5 choices**.

3. **Third Person's Choice**:
   - Once the second person has chosen a seat, only 4 seats remain. So, the third person has **4 choices**.

4. **Fourth Person's Choice**:
   - After the third person is seated, only 3 seats remain. Hence, the fourth person has **3 choices**.

### Total Number of Ways:
To find the total number of ways the 4 persons can take their places, we multiply all the choices:
\[
6 \times 5 \times 4 \times 3 = 360
\]

This is essentially the **permutation** of selecting 4 seats from 6, denoted as \( P(6, 4) \), which is calculated as:
\[
P(6, 4) = \frac{6!}{(6-4)!} = 6 \times 5 \times 4 \times 3 = 360
\]

Thus, the total number of ways the 4 persons can be seated is **360**.

---

Would you like any further details or clarifications on this? Here are some related questions for practice:

1. What if there were 5 persons and 7 vacant seats?
2. How many ways can 3 persons sit in 5 vacant seats?
3. What if all 6 seats must be filled by 4 people?
4. What is the difference between permutations and combinations?
5. How would the calculation change if seating order didn't matter?

**Tip:** When order matters, you use permutations; when order doesn't matter, you use combinations.

Four persons enter a bus in which there are 6 vacant seats. In how many ways can they take their places?

Explore how to solve a combinatorial problem where four people enter a bus with 6 vacant seats and need to take their places. This step-by-step guide explains the use of permutations to calculate the total number of ways they can be seated, using the formula P(6, 4). Suitable for 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