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Question

Four married couples have bought 8 seats in the same row for a concert.

In how many different ways can they be seated

(a) with no restrictions?

(b) if each couple is to seat together?

(c) if all the men sit together to the right of all the women?


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Solution

Step 1:Finding the number of ways four married couple seated with no restriction:

Given,

Total number of people 8.

Total number of seats 8

If there is no restriction then

First seat have 8 possible way.n [since eight persons are there]

Second seat have 7possible way. n-1[since there is only seven person already one person is have seat]

Third seat have 6 possible waysn-2[since there is only six person already two person is have seat]

likewise

seventh seat 2 ways n-6[since there is only two person already six person is have seat]

Eighth seat have 1 way.n-7[since there is only one person already seven person is have seat]

So,

The number of ways be 8×7×6×5×4×3×2×1=40320 ways.

Alter:

We know that,

If there is no restriction in seating arrangement the number of the possible ways =n!

Here n=8

So,

The number of ways be 8!=8×7×6×5×4×3×2×1=40320 ways.

Hence, the number of ways four married couples have bought 8 seats in the same row for a concert with no restriction is 40320 ways.

Step 2: Finding the number of ways four married couples have bought 8 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 =n!

So,

The four married couples can be arranged in 4!=4×3×2×1=24ways

Each pair has 2! ways.[first seat placed for men and second seat for women or first seat placed for women and second seat for men]

So 4 pairs has 2!×2!×2!×2!

So the total number of ways =4!×2!×2!×2!×2!

=4!×2!×2!×2!×2!=24×2×1×2×1×2×1×2×1=384

Hence, the number of ways four married couples have bought 8 seats in the same row for a concert if each couple is to seat together is 384 ways.

Step 3: Finding the number of ways four married couples have bought 8 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 4 men and four 4 women.

The number of arrangement of these 4 men is 4!=4×3×2×1=24 ways [If there is no restriction in seating arrangement the number of the possible ways =n!]

The number of arrangement of these 4 women is 4!=4×3×2×1=24 ways

if all the men sit together to the right of all the women is 1!.

So the total number of ways =4!×4!×1!

=4×3×2×1×4×3×2×1×1=24×24×1=576

Hence, the number of ways four married couples have bought 8 seats in the same row for a concert if all the men sit together to the right of all the women is 576 ways.


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