No. of ways to seat round a table (numbered seats)

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  • Thread starter Punch
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Everything else is correct.In summary, the number of possible arrangements for two families, two men, and two women seated at a round table with numbered seats, where the members of the same family are seated together, is 43,200. This can be calculated by considering the families as separate groups, with 3! possible orders each, and then accounting for the additional choices for seating once the first family is seated.
  • #1
Punch
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Two families are at a party. The first family consists of a man and both his parents while the second familly consists of a woman and both her parents. The two families sit at a round table with two other men and two other women. Find the number of possible arrangements if the members of the same family are seated together and the seats are numbered.

What I did was to consider the 2 families, the 2 woman and 2man as 6 groups of people.
6!(3!)(3!)=25920
but correct answer is 43200
 
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  • #2
If we number the seats 1,2,3,...,10 . Note that, for example, seats 10, 1 and 2 are consecutive seats because we are working with a round table!

So you have to consider the cases in which seats 10 and 1 correspond to the same family too.
 
  • #3
First,i am ignoring the numbers on the seat,

this is a round combination

So, formula is (n-1)!
no.of.ways is 5!(3!)(3!)= 4320

Now the seat are numbered,
then i can more these combinations 1 seats,2seata,...9 seats apart from the original one

so,number of ways is 43,200
 
  • #4
Hello, Punch!

Two families are at a party.
The first family consists of a man and both his parents
. . while the second familly consists of a woman and both her parents.
The two families sit at a round table with two other men and two other women.
Find the number of possible arrangements if the members of the same family
. . are seated together and the seats are numbered.

Answer: 43,200

Duct-tape the families together.

We have: .$\text{(M, P, P)}$ . . . and they have $3!$ possible orders.
We have: .$\text{(W, P, P)}$ . . . and they have $3!$ possible orders.

We also have: .$m,\:m,\:w,\:w$$\text{M}$ has a choice of $10$ seats.
When he is seated, he and his family occupy three seats.
Among the remaining seven seats, $\text{(W, P, P)}$ has $5$ choices for seating.
. . (Think about it.)
Then the remaining four people can be seated in $4!$ ways.Therefore: .$(3!)(3!)(10)(5)(4!) \:=\:43,200$ arrangements.
 
  • #5
grgrsanjay said:
First,i am ignoring the numbers on the seat,

this is a round combination

So, formula is (n-1)!
no.of.ways is 5!(3!)(3!)= 4320

Now the seat are numbered,
then i can more these combinations 1 seats,2seata,...9 seats apart from the original one

so,number of ways is 43,200

I Wanted to know whether my logic holds good for every similar problem??
 
  • #6
grgrsanjay said:
I Wanted to know whether my logic holds good for every similar problem??
Your logic is correct. But why complicate matters?
Once the seats are numbered, we no longer have a circular table.
So there is no need for that.
 

FAQ: No. of ways to seat round a table (numbered seats)

What is the formula for calculating the number of ways to seat round a table with numbered seats?

The formula is n! / n, where n is the number of seats.

How does the number of people seated affect the number of ways to seat round a table?

The number of people seated does not affect the number of ways to seat round a table, as long as there are enough seats for each person.

Can you explain the concept of circular permutations in relation to seating arrangements at a round table?

Circular permutations refer to arrangements that have no distinct starting point or end point. In the case of seating arrangements at a round table, the circular nature of the table means that each seat can be considered as the starting point, resulting in circular permutations.

Is there a difference between seating arrangements with and without numbered seats?

Yes, there is a difference. Seating arrangements with numbered seats have a specific order in which the people are seated, whereas arrangements without numbered seats do not have a set order.

How does the number of seats affect the total number of possible seating arrangements?

The number of seats directly affects the total number of possible seating arrangements. The more seats there are, the larger the number of possible arrangements. This is because each additional seat adds another person to the arrangement, increasing the total number of combinations.

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