If three persons enter a bus in which therw are ten vacant seats fins in how many ways they can sit

To find the number of ways three persons can sit in a bus with ten vacant seats, you need to consider the number of combinations in which three seats can be chosen out of ten, and then the number of permutations of the three persons in those three seats.

1. First, choose 3 seats out of 10. This can be done in \( \binom{10}{3} \) ways, which is the number of combinations of 10 items taken 3 at a time.

\[
\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

2. Next, we need to arrange 3 persons in those 3 seats. This can be done in \( 3! \) ways, which is the number of permutations of 3 items.

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

3. Therefore, the total number of ways three persons can sit in ten vacant seats is:

\[
\binom{10}{3} \times 3! = 120 \times 6 = 720
\]

So, there are 720 different ways for three persons to sit in a bus with ten vacant seats.

Would you like more details or have any questions?

Here are some more questions you might be interested in:

1. How do you find the number of combinations of any n items taken r at a time?
2. What is the difference between permutations and combinations?
3. How do you calculate the factorial of a number?
4. What are the applications of permutations and combinations in real life?
5. Can you provide examples of permutation problems involving different contexts?

**Tip:** When solving problems involving permutations and combinations, always start by determining whether the order of selection matters (permutations) or does not matter (combinations).

Discover how to calculate the number of ways three persons can sit in a bus with ten vacant seats using combinations and permutations. This problem involves understanding combinatorics, permutations, and combinations, 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