Math Problem Statement
there are 11 seats in a row. in how many ways 6 people can be seated if 3 particular people must sit next to each other
Solution
We can think of the three particular people who must sit next to each other as forming a single "block" or group. This reduces the number of units to arrange. Here's how we can break down the problem:
-
Treat the 3 particular people as 1 block: So instead of arranging 6 people in 11 seats, we are arranging 4 units — this block (containing the 3 people) and the 3 other individuals.
-
Arrange the 4 units (block + 3 individuals): The 4 units can be arranged in the 11 available seats. Out of the 11 seats, we need to choose 4 positions where the units will be placed. The number of ways to do this is given by a combination formula: So, there are 330 ways to select 4 seats.
-
Arrange the 3 people within the block: The 3 particular people within the block can be arranged among themselves in (3 factorial) ways. That is:
-
Arrange the remaining 3 individuals: The remaining 3 individuals can be arranged in the remaining 3 seats. This can be done in ways:
-
Final Calculation: To find the total number of ways, multiply all of these values together:
Thus, the total number of ways to seat the 6 people such that the 3 particular people sit next to each other is 11,880.
Would you like a deeper explanation, or do you have any specific questions?
Here are 5 related questions to consider:
- How would the solution change if the row had more or fewer seats?
- What if there were more than 3 particular people who must sit together?
- How many ways could 6 people be seated without any restrictions on seating?
- How does the answer change if two specific people must not sit next to each other?
- Can this method be generalized for different numbers of seats and people?
Tip: When particular people must sit together, treat them as a block first to simplify the arrangement process.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Combinatorics
Factorial
Permutation and Combination
Formulas
Combination formula: \binom{n}{r} = \frac{n!}{r!(n-r)!}
Factorial: n! = n × (n-1) × ... × 1
Theorems
Basic principles of combinatorics
Suitable Grade Level
Grades 9-12
Related Recommendation
Combinatorics Problem: Arranging Friends in Aisle and Non-Aisle Seats
Combinatorics Problem: Circular Seating Arrangements with Group Constraints
Arrange 5 Men, 4 Women, and 3 Children with Men Sitting Together
Permutations of 5 Students with Restrictions: Beth Not Standing Next to Dan
Combinatorial Seating Arrangements for Two Rows of Seats