# Probability problem (#of ways to seat a couple) *edit: solved*

• semidevil
In summary: So the answer is 120 \cdot 32 = 3840.In summary, there are 3840 ways to seat 10 people, consisting of 5 couples, in a row of seats if all couples are to get adjacent seats. This can be calculated by finding the number of ways to distribute 5 two-seat blocks and then accounting for the permutations within each block.
semidevil
update:
*edit: solved! I counted everything twice.

## Homework Statement

How many ways are there to seat 10 people, consisting of 5 couples, in a row
of seats (10 seats wide) if all couples are to get adjacent seats?

## The Attempt at a Solution

I'm trying to look calculate it at a couple level.

10 seats available, 5 couples
_ _ _ _ _ _ _ _ _ _

first couple has 9 choices to choose from They can't choose seat # 10 or else they will not sit next to each other. they can also swap seats, so that will be 9 x 2 = 18 choices.

second couple will have 7 seats to choose from. They can trade seats, so that will be 7 x 2 = 14

third couple has 5 seats, trade seats an yield 10.

fourth couple has 3 seats, = 6

5th couple has 2 seats left, they can trade once, so that is 2 possibilities

I did 18 x 14 x 10 x 6 x 2 = 30,240. answer key says 3840 though?

Last edited:
Since you have already found the correct answer, I will share another way to get there.
As each couple has to sit next to each other, you basically have 5 two-seat blocks to distribute the 5 couples over. The number of ways in which this can be done is just 5! because every permutation of the couples will give a different assignment of couples to two-seat blocks. Then within each block you can swap the two persons, so you get $5! \cdot 2^5$ possible ways.

## 1. How many ways can a couple be seated at a table with 8 chairs?

There are 8! = 40,320 ways to seat a couple at a table with 8 chairs, assuming that the couple is not required to sit next to each other.

## 2. What is the probability that a couple will be seated next to each other at a table with 8 chairs?

The probability of a couple being seated next to each other at a table with 8 chairs is 2/8 = 1/4 or 25%. This is because there are 2 possible ways for the couple to be seated next to each other (either the man sits next to the woman or the woman sits next to the man) out of a total of 8 possible seats.

## 3. How many ways can a couple be seated at a table with 6 chairs if there are 3 other people already sitting?

There are 6! = 720 ways to seat a couple at a table with 6 chairs if there are already 3 other people sitting. This is because there are 6 remaining seats for the couple to choose from, and the order in which the other 3 people are already seated does not matter.

## 4. Is there a difference in the number of ways to seat a couple at a rectangular table compared to a circular table?

Yes, there is a difference. The number of ways to seat a couple at a rectangular table with n chairs is (n-1)!/2, while the number of ways to seat a couple at a circular table with n chairs is (n-1)!.

## 5. Can you use the same formula for calculating the number of ways to seat a group of people as you do for calculating the number of ways to seat a couple?

Yes, the same formula can be used for calculating the number of ways to seat any group of people. The formula is n!, where n is the number of people in the group.

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