Solution:
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.
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PHOTOGRAPHY A photographer is taking a picture of a bride and groom together with 6 attendants. How many ways can he arrange the 8 people in a row if the bride and groom stand in the middle?
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(a) Considering five couples as five special objects, we have 5! = 5*4*3*2*1 = 120 permutations of these objects. Next, we have 2 possible permutations (A,B) --> (B,A) inside each of 5 pairs. These permutations are independent --- so, there are
- Introduction to Permutations
- PROOF of the formula on the number of Permutations
- Simple and simplest problems on permutations
- Special type permutations problems
- Problems on Permutations with restrictions
- OVERVIEW of lessons on Permutations and Combinations in this site. Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations".
Save the link to this textbook together with its description Free of charge online textbook in ALGEBRA-II //www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson into your archive and use when it is needed.7. Six couples want to have their pictures taken. In how many ways can they arrange themselves in a row i
couples must stay together?
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8. How many five-digit numbers can be formed from the numbers
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Given: n = r = 6 couples = 12 people
Formula: P(n,r) = n!/(n-r)!
Solution:
P(12,12) = 12!/(12-12)!
P(12,12) = 12!
P(12,12) = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 479,001,600 ways