Thủ Thuật Hướng dẫn How many ways can 4 married couples seat themselves around a circular table if spouses sit opposite each other? Chi Tiết
Cao Thị Phương Thảo đang tìm kiếm từ khóa How many ways can 4 married couples seat themselves around a circular table if spouses sit opposite each other? được Cập Nhật vào lúc : 2022-09-20 18:10:08 . Với phương châm chia sẻ Bí kíp Hướng dẫn trong nội dung bài viết một cách Chi Tiết Mới Nhất. Nếu sau khi tham khảo tài liệu vẫn ko hiểu thì hoàn toàn có thể lại Comment ở cuối bài để Admin lý giải và hướng dẫn lại nha.In how many ways can 4 married couples (total of 8 people) be seated in a row if: (a) there are no restrictions on the seating arrangement? (b) persons A and B must sit next to each other?(c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?
Solution:
Nội dung chính- In how many ways can 4 married couples (total of 8 people) be seated in a row if: (a) there are no restrictions on the seating arrangement? (b) persons A and B must sit next to each other?(c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?In how many ways can 4 married couples (total of 8 people) be seated in a row if: (a) there are no restrictions on the seating arrangement? (b) persons A and B must sit next to each other?(c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?
In how many ways 4 married couples can be seated round a table if no husband and wife as well as two no two men are to be in adjacent seats.A. 384B. 14C. 24D. 36
How many ways can 6 couples be seated in a circular table?What is the formula for circular permutation?How many ways are there to arrange the first five couples in a round table?How many ways can you Permute 5 objects in a circle?
Given, there are 4 married couples
Total number of people = 8
we have to find how many ways can 4 married couples be seated in a row
a) if there are no restrictions on seating arrangement
The number of possible ways = n!
Here, n! = 8!
= 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40320 ways.
Therefore, there are 40320 ways that the people can be seated when there is no restriction on the seating arrangement.
b) if persons A and B must sit next to each other
The possible ways = 7! × 2!
= (7 × 6 × 5 × 4 × 3 × 2 × 1) × (2 × 1)
= 5040 × 2
= 10080
Therefore, if persons A and B must sit next to each other there are 10080 ways of seating arrangement.
c) if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other
This implies the restriction of having persons of opposite sex next to each other.
We can have any one of the 8 persons in the first position.
So, both man and woman can be arranged without having same next to
each other = 8 × 4 × 3 × 3 × 2 × 2 × 1 × 1
= 1152 ways
Therefore, if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other, there are 1152 ways of seating arrangement.
d) if there are 5 men and they must sit next to one another
There are 5 men and 3 women. 5 men sit next to one another.
The possible number of ways = 5! × 4!
= (5 × 4 × 3 × 2 × 1) × (4 × 3 × 2 × 1)
= 120 × 24
= 2880 ways
Therefore, if there are 5 men and they must sit next to one another, there are 2880 ways of seating arrangement.
e) if there are 4 married couples and each couple must sit together
4 pairs of couples can be arranged in 4!
Each couple can be arranged in 2! Ways.
The possible number of ways = 2! × 2! × 2! × 2!
= (2× 1) × (2 × 1) × (2 × 1) × (2 × 1) × (4 × 3 × 2 × 1)
= 2 × 2 × 2 × 2 × 24
= 16 × 24
= 384 ways
Therefore, if there are 4 married couples sitting together, there are 384 ways of seating arrangement.
In how many ways can 4 married couples (total of 8 people) be seated in a row if: (a) there are no restrictions on the seating arrangement? (b) persons A and B must sit next to each other?(c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?
Summary:
The possible number of ways can 4 married couples (total of 8 people) be seated in a row if:
(a) there are no restrictions on the seating arrangement is 40320 ways
(b) persons A and B must sit next to each other is 10080 ways
(c) There are 4 men and 4 women and no 2 men or 2 women can sit next to each other in 1152 ways.
(d) there are 5 men and they must sit next to one another is 2880 ways
(e) there are 4 married couples and each couple must sit together in 384 ways.
In how many ways 4 married couples can be seated round a table if no husband and wife as well as two no two men are to be in adjacent seats.A. 384B. 14C. 24D. 36
Answer
Verified
Hint: To solve this problem we have to know about the concept of permutations and combinations. But here a simple concept is used. In any given word, the number of ways we can arrange the word by jumbling the letters is the number of letters present in the word factorial. Here factorial of any number is the product of that number and all the numbers less than that number till 1.
$ Rightarrow n! = n(n - 1)(n - 2).......1$
Complete step-by-step solution:
Given there
are four pairs of married couples, in which each couple contains one male and one female.
In a married couple, the male is called the husband and the female is the wife.
We have to arrange them in such a way that no husband and wife of the same marriage couple can sit beside each other and also men should not be seated beside each other. That means that no two men can be seated beside each other.
Here there are 4 men and 4 women, which makes 4 married couples.
Consider the 4 married
couples be M1F1, M2F2, M3F3, M4F4. Where M stands for male and F stands for female.
Now no two men can be seated beside each other but these 4 men can be arranged in $4!$ways.
That is placing these 4 men alternatively, which can be arranged in $4!$ ways = $24$ ways.
$because 4! = 4 times 3 times 2 times 1$
$ Rightarrow 4! = 24$
Let’s say we placed the 4 men like this:
M1_M2_M3_M4_.
Now as no couple can sit together so each blank has a choice of only 2 women.
Hence
for each blank only 2 choices can be placed, as there are 4 blanks, the no. of ways in which the women are arranged as: $2 times 2 times 2 times 2$
Thus the no. of ways 4 married couples can be seated round a table, where no husband and wife as well as two no two men are to be in adjacent seats, is given by:
[ Rightarrow 24 times 2 times 2 times 2 times 2]
[ Rightarrow 24 times 4 times 4]
[ Rightarrow 24 times 16]
[ Rightarrow 384]
$therefore $The no. of
ways in which the 4 married couples can be seated round a table where no husband and wife as well as two no two men are seated beside is 384 ways.
The arrangement can be done in 384 ways.
Option A is the correct answer.
Note: Here while solving this problem one thing we have to understand is that why each blank is filled with only 2 women, for example consider the first blank, in that F1 or F2 can’t be placed as M1 is on the left of the blank which makes M1F1 a couple, and M2 is on the right of the blank which makes F2M2 a couple. Hence we can only place either F3 or F4, hence only 2 ways.
How many ways can 6 couples be seated in a circular table?
=720. I think you should take some time to understand the question before trying to answer it.What is the formula for circular permutation?
1 What is the formula for circular permutations? If clockwise and anti-clockwise orders are different, then a total number of circular permutations is given by P n = ( n − 1 ) !How many ways are there to arrange the first five couples in a round table?
In how many different ways can five people be seated a circular table? So the answer is 24.How many ways can you Permute 5 objects in a circle?
Coming back to the question – In how many ways can 5 distinct objects be arranged in a circle? Now there are 5! or 120 different linear arrangements possible. Therefore the number of circular arrangements will be 5! ÷ 5. Tải thêm tài liệu liên quan đến nội dung bài viết How many ways can 4 married couples seat themselves around a circular table if spouses sit opposite each other?