Bunuel wrote:
In how many ways can 5 men and 5 women be arranged in a circle if the men are separate and if two particular women must not be next to a particular men?
A. 432
B. 864
C. 1296
D. 1440
E. 2880
Solution:If the men must be separated and since there are an equal number of men and women, then there must be one man sitting between every two women and likewise, one woman sitting between every two men.
Recall that the number of permutations of n objects in a circular fashion is (n -1)!. Therefore, if the five men were to be seated first, there are (5 - 1)! = 4! = 24 ways. Then the five women can be seated in 5! = 120 ways. Therefore, there are a total of 24 x 120 = 2880 ways.
Now, let M be the particular man and A, B, C, D, and E be the five women. If there are no restrictions, we could have:
AMB, AMC, AMD, AME, BMA, BMC, BMD, BME, CMA, CMB,
CMD,
CME, DMA, DMB,
DMC,
DME, EMA, EMB,
EMC,
EMDWe see that there are 20 seating arrangements. However, let’s say A and B are the two particular women that M can’t sit next to; then we only have 6 seating arrangements (in bold). In other words, we only have 6/20 of the 2880 sitting arrangements if a particular man can’t sit next to two particular women. Therefore, we only have 6/20 x 2880 = 864 possible seating arrangements.
Answer: B