10 persons are to be arranged on a round table. If two particular persons among them are not to be side by side, then the total number of arrangements is?
Understand the Problem
The question asks for the total number of arrangements for 10 persons seated at a round table, with the condition that two specific persons cannot sit next to each other. This typically involves combinatorial mathematics to solve.
Answer
The total number of valid arrangements is \( 7 \times 8! \).
Answer for screen readers
The total number of arrangements is ( 7 \times 8! ).
Steps to Solve
- Calculate Total Arrangements Without Restrictions
For arranging $n$ people around a round table, the number of arrangements is given by $(n-1)!$. Therefore, for 10 people: $$ T = (10 - 1)! = 9! $$
- Calculate Arrangements Where the Two Persons Are Together
Treat the two specific persons as one unit or block. This results in 9 units to arrange (the 2 persons as one unit and the other 8 individuals). The arrangements for these 9 units is: $$ A = (9 - 1)! = 8! $$
- Calculate the Arrangements Within the Block
Since the two specific persons can be arranged amongst themselves in $2!$ ways, the total arrangements considering this block is: $$ B = 8! \times 2! $$
- Calculate Total Valid Arrangements
To find the total arrangements where the two specific persons are not next to each other, we subtract the arrangements where they are together from the total arrangements without restrictions: $$ \text{Valid Arrangements} = 9! - (8! \times 2!) $$
- Compute the Final Answer
Simplifying the previous equation: $$ \text{Valid Arrangements} = 9! - 2 \times 8! = (9 - 2) \times 8! = 7 \times 8! $$
The total number of arrangements is ( 7 \times 8! ).
More Information
In arranging people around a circular table, the method of treating some persons as blocks is a common combinatorial technique. Since the arrangements are circular, we reduce the count by fixing one person's position.
Tips
- Forgetting that arrangements in a circle reduce the number of distinct configurations by fixing one person can lead to counting errors.
- Not accounting for the arrangements of the two people being considered together can result in incorrect totals.
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