A dinner party is attended by four men and four women. How many unique ways can

Question

A dinner party is attended by four men and four women. How many unique ways can the eight people sit around the table? How many unique ways can the people sit around the table with men and women alternating seats?

Using pictures and words, please describe/show/prove why a plausble answer to the above questions is the following:

ROUND TABLE
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part a)
the formula for unrestricted seating for a round table is (n-1)!
because there is no "defined starting point",
so wherever the 1st person sits is taken to be the defined starting point

# of ways = 7! = 5040
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part b)
imagine there are only 4 chairs to start with.
the 1st lady can sit anywhere, the remaining 3 can sit in 3! = 6 ways

now insert 4 other chairs so that men-women alternate.
the men can sit in 4! = 24 ways in relation to the women

# of ways = 3!4! = 144 ways

Explanation / Answer

a)Normally for unrestricted seating in a row number of ways will be n! . But in case of round table some cases repeat as the first person is the neighbour of last person as well and vice versa. That's why that case has to remove for all n persons . Hence , possible ways = (n-1)! .

b)In this case , there is a restriction that men and women should sit alternate. That's why if the 1st lady can sit anywhere, the remaining (n-1) can sit in (n/2-1)! ways. Now the men can sit in lest (n/2)! .

So total ways : (n/2-1)! * (n/2)!

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