a) If there are no seating restrictions, there are 720 different ways that three couples can be seated.
b) If each couple sits together, there are 48 different arrangements for three couples to be seated.
What is Fundamental counting principle?
If an event can happen in m different ways and another event can happen in n different ways, then the sum of the occurrences of the two events is m x n.
Given:
Three couples have reserved seats in a row for a concert.
(a) There are 3 couples, so 6 people must be seated in seats in row.
It is permutation of 6 elements, therefore, there are
6!=720 ways three couples can be seated when there is no seating restriction.
(b) We want that each couple sits together.
First we will find the number of permutations of 3 couples:
There are 3!=6 permutations of couple. (they can be ordered in 6 different ways).
Now we will find the number of permutations of each couple:
There are 2!=2 permutations of 2 people that makes a couple.
Therefore, by Fundamental Counting Principle, there are
3! ⋅ 2! ⋅ 2! ⋅ 2! = 6 ⋅ 2 ⋅ 2 ⋅ 2 = 48
ways three couples can be seated when each couple sits together.
Hence,
a) If there are no seating restrictions, there are 720 different ways that three couples can be seated.
b) If each couple sits together, there are 48 different arrangements for three couples to be seated.
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Complete question:
Three couples have reserved seats in a row for a concert. In how many different ways can they be seated when (a) there are no seating restrictions? (b) each couple sits together?