3. If three couples, a couple being respectively the lady and her gentleman date

Question

3. If three couples, a couple being respectively the lady and her gentleman date, have reserved six adjacent seats at the dinner theater, determine how many ways they can be seated given the ensuing.   [Hint: Factorial; address the special cases first]

3.1 Two gentlemen are identical twins and are dressed identically and not differentiable

3.2 Each gentleman sits adjacent to a lady other than his date       

3.3 Persons of the same sex sit in the first two seats

3.4 Two couples each sit with their date

3.5 One couple is already seated

3.6 Couples sit together but one couple will not sit adjacent to another couple

Explanation / Answer

Solution

There are totally 6 persons. Just for convenience of explaining and presentation, let the couples be (G1, L1), (G2, L2) and (G3, L3) where Li is the date of Gi.

Back-up Theory

1. n distinct things can arranged among themselves in n! ways, where n! = n(n - 1)(n - 1)….1.

2. Of the n things, if p are identical, the above becomes (n!)/(p!)

Part (3.1)

This situation is the same as arranging 6 things where 2 things are identical.

Vide (2) of Back-up Theory, total number of possible seating arrangements = (6!)/(2!)

= 360 ANSWER

Part (3.2)

This is ideally solved by actual enumeration. The possible seating arrangements are:

G3L2G1L3G2L1

L2G1L3G2L1G3

G3L1G2L3G1L2

L1G2L3G1L2G3

G2L1G3L2G1L3

L1G3L2G1L3G2

Thus, total number of possible seating arrangements = 6 ANSWER

Part (3.4)

Out of 3 couples, two couples sitting with their dates is possible in 3 ways. In each of these 3 ways, the couples can interchange their positions as a couple in 2 ways and in each of these cases, within a couple. G and L can interchange their positions in 2 ways. Thus, there are 12 ways two couples can sit with their dates. Now, of the remaining couple, say G and L, one of them can sit in between two couples or on either side of the two couples. Thus there are 3 possibilities and for each of these 3 possibilities, the other can occupy either of the remaining 2 positions. Thus, total number of possible seating arrangements = 72 ANSWER

Part (3.5)

If one couple is already seated, there are 4 seats remaining to be occupied by 4 persons. This is possible in 4! = 24 ways.

Thus, total number of possible seating arrangements = 74 ANSWER

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