Understanding the Problem
To seat 4 Americans and 4 English men at a round table such that no two Americans sit next to each other, we will utilize combinatorial principles.
Step 1: Fixing Positions for English Men
- When arranging people at a round table, we can fix one person's position to eliminate identical rotations.
- Fix one English man’s position. This leaves us with 3 English men to arrange.
Step 2: Arranging English Men
- The remaining 3 English men can be seated in the remaining seats.
- The number of ways to arrange 3 English men is:
- 3! (factorial) = 6 ways.
Step 3: Identifying Available Spots for Americans
- After seating the English men, there will be 4 gaps created between them (including the gap between the last and the first English man).
- This ensures that no two Americans are seated next to each other.
Step 4: Placing the Americans
- We need to choose 4 out of the 4 available gaps to seat the Americans.
- The number of ways to arrange the 4 Americans in these gaps is:
- 4! = 24 ways.
Step 5: Calculating Total Arrangements
- The total number of arrangements is the product of the arrangements of English men and Americans:
- Total arrangements = (Arrangement of English men) × (Arrangement of Americans) = 6 × 24 = 144.
Final Result
The total number of ways to seat 4 Americans and 4 English men at a round table such that no two Americans sit next to each other is 144.