There are 6 gentlemen and 5 ladies to dine at a round table. In how many ways can they seat themselves so that no two ladies are seated together.Bunuel
already gave very clear answers, but because antoxavier
originally asked about this through Magoosh
, I want to explain a little further.
Arrangements in circles are tricky in that we have to be careful not to count the same arrangement multiple times. For example, with five people, the arrangements ABCDE and DEABC look different but are actually the same --- every person has the same people on each side. We have just rotated the arrangement around the circle, but we don't count that as a "new" arrangement.
One way to make sure we don't overcount is to pick one distinctive feature of the arrangement and orient everything from that. Here, we are particularly fortunate. If we had five males and five females, they could simply be alternating. The fact that we have six
males and five females means there is exactly one place around the table where two guys are sitting next to each other.
Starting from the two guys sitting next to each other, we have M M F M F M F M F M F
Those six guys can be rearranged in those six slots, and no arrangement will be counted twice --- the clump of two guys breaks the symmetry, so each arrangement is unique. That's a permutation of six individuals, 6P6 = 720.
Then the females be arranged in those five slots --- again, since the two guy group breaks the symmetry, any arrangement of the women will be unique. That's 5P5 = 120.
By the Fundamental Counting Principle
, we multiply these to get the total number of seating arrangements.
Total number of possibilities = (720)*(120) = 86400, answer C
Notice, this question would become considerably more difficult if we merely changed it to seven
guys and five woman --- now twelves total seats, and it's still the case that no two women can sit together.
Now, we could have a clump of three
guys sitting together, and then all the other guys alternating with women ---- OR
, we could have two clumps of two guys next to each other, and these two clumps could be distributed at different positions around the table --- we would have to count all those arrangements, and then for each one, figure out all the permutations of males & females. A much trickier question.
Does all this make sense?
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