Question

=

Eight people (four married couples) arrange themselves randomly in eight consecutive seats in a row. Complete
parts (a) and (b) below. (Hint: The denominator of the probability fraction will be 8!= 40,320, the total number of ways
to arrange eight items.)
The women will be in four adjacent seats.
(a) Determine the number of ways in which the described event can occur.
The event can occur in
ways.
(Type a whole number.)
(b) Determine the probability of the event.
The probability is
(Type an integer or a simplified fraction.)

Answer 1

Explanation:

(a) To determine the number of ways in which this event can occur, you can treat each couple as a single entity. There are four couples, so you have 4 entities (2 people per entity). The number of ways to arrange these 4 entities in a row is 4!.

Within each couple, there are 2 ways to arrange the husband and wife, so you need to multiply by 2 for each couple.

Now, within each couple, there are 2 ways to arrange the husband and wife, so you need to multiply by 2 for each couple.

So, the total number of ways to arrange the people with the women in four adjacent seats is:

4

!

×

(

2

4

)

=

24

×

16

=

384

4!×(2

4

)=24×16=384 ways.

(b) To determine the probability of this event, you'll divide the number of favorable outcomes (384 ways from part a) by the total number of possible outcomes, which is 8! (40,320 ways).

The probability is:

384

40

,

320

=

384

40

,

320

=

3

320

40,320

384

​

=

40,320

384

​

=

320

3

​

So, the probability of the event is

3

320

320

3

​

.