Four married couples have bought seats in the same row for a concert.
In how many different ways can they be seated
with no restrictions?
if each couple is to seat together?
if all the men sit together to the right of all the women?
Step 1:Finding the number of ways four married couple seated with no restriction:
Given,
Total number of people .
Total number of seats
If there is no restriction then
First seat have possible way. [since eight persons are there]
Second seat have possible way. [since there is only seven person already one person is have seat]
Third seat have possible ways[since there is only six person already two person is have seat]
likewise
seventh seat ways [since there is only two person already six person is have seat]
Eighth seat have way.[since there is only one person already seven person is have seat]
So,
The number of ways be ways.
Alter:
We know that,
If there is no restriction in seating arrangement the number of the possible ways
Here
So,
The number of ways be ways.
Hence, the number of ways four married couples have bought seats in the same row for a concert with no restriction is ways.
Step 2: Finding the number of ways four married couples have bought seats in the same row for a concert if each couple is to seat together:
If there are four married couples.
As a couple they are calculated as one . Couples have no restriction
We know that,
If there is no restriction in seating arrangement the number of the possible ways
So,
The four married couples can be arranged in ways
Each pair has ways.[first seat placed for men and second seat for women or first seat placed for women and second seat for men]
So pairs has
So the total number of ways
Hence, the number of ways four married couples have bought seats in the same row for a concert if each couple is to seat together is ways.
Step 3: Finding the number of ways four married couples have bought seats in the same row for a concert if all the men sit together to the right of all the women:
Four married couples means men and four women.
The number of arrangement of these men is ways [If there is no restriction in seating arrangement the number of the possible ways ]
The number of arrangement of these women is ways
if all the men sit together to the right of all the women is .
So the total number of ways
Hence, the number of ways four married couples have bought seats in the same row for a concert if all the men sit together to the right of all the women is ways.