https://file.aiquickdraw.com/mathbot/file/17273618411995xle9k8v.jpg

The problem asks how four persons can take their places on a bus with six vacant seats. This is a combinatorics problem involving permutations.

### Step-by-step solution:
1. **First person's choice**: The first person has **6 vacant seats** to choose from, so there are 6 possible ways for the first person to sit.
   
2. **Second person's choice**: Once the first person has chosen a seat, only 5 seats remain vacant. Hence, the second person has **5 choices**.

3. **Third person's choice**: Similarly, the third person has **4 choices** (since two seats are already taken).

4. **Fourth person's choice**: The last person has **3 choices** (since three seats are already taken).

### Permutation Calculation:
To find the total number of ways the 4 people can be seated in 6 vacant seats, you multiply the choices at each step:

\[
6 \times 5 \times 4 \times 3 = 360
\]

Thus, the total number of ways the four persons can seat themselves is **360**.

This method uses the **Permutation formula** \( P(6, 4) \), which is:

\[
P(6, 4) = \frac{6!}{(6-4)!} = 6 \times 5 \times 4 \times 3 = 360
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. What if there were 5 people and still 6 vacant seats?
2. How many ways can these 4 persons sit if the number of vacant seats was 8?
3. What is the difference between permutations and combinations?
4. How would the result change if the seats were distinguishable by color or label?
5. What if 2 of the 4 persons must sit together?

**Tip:** In permutations, the order of selection matters, while in combinations it doesn't!

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

Explore how to calculate the number of ways 4 people can take their places in 6 vacant bus seats. This problem demonstrates key permutation concepts and uses the formula P(6, 4) to find the solution, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation