How many ways can 6 people be seated in a row if two specific people must sit next to each other?
Practice Questions
1 question
Q1
How many ways can 6 people be seated in a row if two specific people must sit next to each other?
120
720
240
60
Treat the two specific people as one unit. Then, we have 5 units to arrange: 5! = 120. The two people can be arranged in 2! ways. Total = 120 * 2 = 240.
Questions & Step-by-step Solutions
1 item
Q
Q: How many ways can 6 people be seated in a row if two specific people must sit next to each other?
Solution: Treat the two specific people as one unit. Then, we have 5 units to arrange: 5! = 120. The two people can be arranged in 2! ways. Total = 120 * 2 = 240.
Steps: 8
Step 1: Identify the two specific people who must sit next to each other. Let's call them Person A and Person B.
Step 2: Treat Person A and Person B as one single unit or block. This means instead of 6 individual people, we now have 5 units to arrange: the block (A and B together) and the other 4 people.
Step 3: Calculate the number of ways to arrange these 5 units. The formula for arranging n units is n!. So, we calculate 5! (which is 5 factorial).
Step 4: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120. This is the number of ways to arrange the 5 units.
Step 5: Now, within the block of Person A and Person B, they can switch places. This means we can arrange them in 2! ways (which is 2 factorial).
Step 6: Calculate 2! = 2 × 1 = 2. This is the number of ways to arrange Person A and Person B within their block.
Step 7: To find the total number of arrangements, multiply the number of ways to arrange the 5 units by the number of ways to arrange Person A and Person B. So, we calculate 120 × 2 = 240.
Step 8: The final answer is that there are 240 different ways to seat the 6 people in a row with Person A and Person B sitting next to each other.
