Question 1061664: in how many ways can10 people be seated in a round table if 3 persons want to sit next to one another?
Found 2 solutions by josmiceli, Edwin McCravy:
Answer by josmiceli(19441)   (Show Source):
Answer by Edwin McCravy(20055)  (Show Source):
You can put this solution on YOUR website!
"Round table" permutations are considered as if the table
and chairs and people were on a rotating platform, like the
horses on a merry-go-round. Though this is not realistic
for table seating, it is nevertheless the accepted way to
assume that "round table" mathematics problems are to be
interpreted.
In building a merry-go-round with a single circle of n
horses, the number of orders of n different colored horses
can be installed on the merry-go-round is (n-1)!
[That's because it would be n! if the merry-go-round could
only remain still, i.e., could not rotate. But each of the
n rotations would be a different one of the n! "still"
arrangements. So each "still" arrangement is counted n
times among the n!, so we divide the n! by n to get (n-1)!
So when n things are arranged in a line, the number of
possible arrangements is n!, but when they are in a circle,
the number of possible arrangements is (n-1)!
So arranging in a straight line, the formula is n!,
and at a round table, it's (n-1)!
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Since 3 people want to sit next to each other,
we can have them sit as a trio in 3! or 6 ways.
For each of the 3! choices for the trio, we have
7 single people plus 1 trio to seat round the table.
That's 8 things
Using the (n-1)! "round table" formula, the answer is
3!(8-1)! = 3!7! = 6(5040) = 30240
Edwin
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