Question 1190125: In how many ways can a family of 8 members be seated at a circular table with only 5 chairs?
Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website!
Because the problem is stated incorrectly, the answer from the other tutor is correct: 0.
8 people can't be seated in 5 chairs.
Correct the statement of the problem:
"In how many ways can 5 members of a family of 8 be seated at a circular table with only 5 chairs?"
pick any one of the 5 seats to be the "1st" seat; there are 8 choices for the family member to sit in that seat
pick any one of the remaining seats to be the "2nd" seat; there are 7 choices for the family member to sit in that seat
pick any one of the remaining seats to be the "3rd" seat; there are 6 choices for the family member to sit in that seat
pick either of the remaining seats to be the "4th" seat; there are 5 choices for the family member to sit in that seat
there is only one remaining seat to be the "5th" seat; there are 4 choices for the family member to sit in that seat
ANSWER (number of ways to seat 5 members of a family of 8 at a circular table with 5 chairs): 8*7*6*5*4 = 6720
In the popular literature, it is likely that the "expected" correct answer is 6720/5 = 1344 -- the reason being that, when seated around a circular table, any one of the 5 chairs can be considered the "first".
However, that is NOT how the problem is stated.
The number of DIFFERENT ORDERS in which 5 members of a family of 8 can be seated around a circular table is 6720/5 = 1344. But the number of DIFFERENT WAYS they can be seated is 6720.
We can demonstrate the difference by considering the simple case of 3 people A, B, and C seated around a circular table:
A A B B C C
B C C B A C C A A B B A
That is 6 different WAYS but only 6/3 = 2 different ORDERS.
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