Question:

A person invites a party of 10 friends at dinner and places so that 4 are on one round table and 6 on the other round table. Total number of ways in which he can arrange the guests is:

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In problems involving round table arrangements, subtract 1 from the number of people to account for rotational symmetry.
Updated On: May 22, 2025
  • \( \frac{10!}{6!} \)
  • \( \frac{10!}{24} \)
  • \( \frac{9!}{24} \)
  • None of these
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The Correct Option is B

Approach Solution - 1

Step 1: {Total number of people} 
The total number of people is \( 10 \), and they are divided into two groups: one group of 4 persons and another group of 6 persons. 
Step 2: {Number of ways to form groups} 
The number of ways to form the two groups is \( \frac{10!}{4!6!} \). 
Step 3: {Arrangements on the round tables} 
For the group of 4 persons, the number of arrangements on a round table is \( (4-1)! = 3! = 6 \). For the group of 6 persons, the number of arrangements on a round table is \( (6-1)! = 5! = 120 \). 
Step 4: {Total ways to arrange the guests} 
The total number of ways to arrange the guests is: \[ \frac{10!}{4!6!} \times 6 \times 120 = \frac{10!}{24}. \] Thus, the total number of ways is \( \frac{10!}{24} \), which matches option (B). 
 

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Approach Solution -2

Step 1: Understand the problem
There are 10 friends to be seated at two round tables:
- One table seats 4 people
- The other table seats 6 people

Step 2: Divide friends into two groups
Number of ways to choose 4 friends out of 10 for the first table:
\[ \binom{10}{4} = 210 \]

Step 3: Arrange friends around each round table
For a round table with \(n\) people, the number of distinct arrangements is \((n-1)!\) because rotations are considered the same.
- For 4 people: arrangements = \( (4-1)! = 3! = 6 \)
- For 6 people: arrangements = \( (6-1)! = 5! = 120 \)

Step 4: Calculate total arrangements
Multiply ways to choose the groups and arrange each group:
\[ 210 \times 6 \times 120 = 210 \times 720 = 151200 \]

Step 5: Express result in factorial form
\[ \frac{10!}{4! \times 6!} \times 3! \times 5! = \frac{10!}{4!6!} \times 6 \times 120 \]
Simplify:
\[ = \frac{10!}{4!6!} \times 720 = \frac{10! \times 720}{4!6!} \] Since \(4! = 24\) and \(6! = 720\), denominator is \(24 \times 720 = 17280\). So:
\[ \frac{10! \times 720}{17280} = \frac{10!}{24} \]

Final Answer: \( \frac{10!}{24} \)
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