Max@Math Revolution wrote:[GMAT math practice question]
There are 5 couples. If they sit on 10 chairs around a round table such that each couple sits side by side, how many possible cases are there?
A. 256
B. 512
C. 768
D. 1,024
E. 1,080
For circular arrangements:
1. Place an element at the table.
2. Count the number of ways to arrange the REMAINING elements.
Once a couple has been placed at the table:
Number of ways to arrange the remaining 4 couples = 4! = 24.
Since the members of each couple can be arranged 2 ways -- and there are 5 couples in total -- the result above must be multiplied by 2*2*2*2*2:
24 * 2� = 768.
The correct answer is
C.
Alternate approach:
Once a person has been placed at the table:
Number of options for the first person's spouse = 2. (To the right or left of the the first person.)
Moving clockwise around the table:
Number of options for the next clockwise seat = 8. (Any of the 8 remaining people.)
Number of options for the next clockwise seat = 1. (Must be the spouse of the person just seated.)
Number of options for the next clockwise seat = 6. (Any of the 6 remaining people.)
Number of options for the next clockwise seat = 1. (Must be the spouse of the person just seated.)
Number of options for the next clockwise seat = 4. (Any of the 4 remaining people.)
Number of options for the next clockwise seat = 1. (Must be the spouse of the person just seated.)
Number of options for the next clockwise seat = 2. (Either of the 2 remaining people.)
Number of options for the last seat= 1. (Only 1 person left.)
To combine the options above, we multiply:
2*8*1*6*1*4*1*2*1 = 768.
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