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There are 2 indian couples, 2 american couple and 1 unmarried person
List- IList-II(I)The total mumber of ways in which they can sit(P) 5760 in a row such that an indian wife and an american wife are always on either side of the unmarried person(Q) 24230(II)The total mumber of ways in which they can sit in a row such that the unmarried man alwaysoccupy the midddle position(R) 40320(III)The total mumber of ways in which they can sitaround a circular table such that indian wife and american wife are on either side of unmarried person(S) 1920(IV)The total number of ways in which they can sit in a row such that all couples sit together(T) 28410

1) Which of the following is only CORRECT combination?

A
I(T)
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B
II(Q)
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C
III(P)
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D
IV(P)
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Solution

The correct option is C III(P)
For case I :
An indian wife and an american wife can be selected in = 2C1× 2C1=4 ways
Now, we need to make arrangements in which both of these indian and american wives sit around the unmarried person. There are two possibility as (IUA),(AUI) (I, U and A represent indian wife, unmarried person and american wife respectively).
we will consider (IUA or AUI) as a one element and there are 6 elements (persons in this case) remain. That can be arranged in 7! ways
So, total number of ways =4×2×7!=8!

For case II
Number of ways in which the unmarried person can be arranged =1
Now, other 8 persons can be arranged in 8! ways.
So, total number of ways = 8!

For case III
This case is same as the case I, but here the arrangement is circular.
We will consider (AUI or IUA) as a one object and there are 6 objects (persons) remain. so, the number of ways of arranging the 7 objects in a circluar table 6!
So, total number of ways =4×2×6!=5760

For case IV
Here, we have to arrange 4 couples and 1 unmarried person. The total number of ways =5!×(2!)4=1920 (a couple can be internally arranged in 2! ways)

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