5 men and 5 women, including Michael and Alice, participated in a match-making session at a restaurant. All participants are to sit in a way such that no two persons of the same gender sit next to each other. How many ways can the participants be arranged if they are seated on both sides of a rectangle table with 5 seats on each side, such that Michael and Alice sit next to each other on the same side? (the seats are labelled).
ANS: 9216
please show steps thanks
ANS: 9216
please show steps thanks
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Let one of the the table configuration be as:
X X X X X
M A X X X
Now
Michael and Alice can sit together on each side of the table together in (5 - 1) = 4 ways. Hence in total they can have 2 * 4 = 8 ways t grab a seat together on either side of the table. They can further be arranged amongst themselves in 2! = 2 ways hence making a total of
8 * 2 = 16 ways in which they can be arranged. Now in all seating configurations created by MA, we can chose the alternate gender combinations for men and women in only 1 way and they can be arranged in that seating configuration in 4! ways for men and another 4! ways for women
hence total arrangement will be
16 * 4! * 4! = 9216
X X X X X
M A X X X
Now
Michael and Alice can sit together on each side of the table together in (5 - 1) = 4 ways. Hence in total they can have 2 * 4 = 8 ways t grab a seat together on either side of the table. They can further be arranged amongst themselves in 2! = 2 ways hence making a total of
8 * 2 = 16 ways in which they can be arranged. Now in all seating configurations created by MA, we can chose the alternate gender combinations for men and women in only 1 way and they can be arranged in that seating configuration in 4! ways for men and another 4! ways for women
hence total arrangement will be
16 * 4! * 4! = 9216