perms (1 Viewer)

[moderntortoisecat]

in how many ways can five writers and five authors be arranged in a circle so that the writers are separated? In how many ways can this be done if 2 particular artists must not sit with one particular writer?

Part a i got (2880) but part b is confusing me, i was considering taking the complementary by no restrictions - 2 artists with one writer but i cant seem to get the answer of 864.

Thanks!

 

[liamkk112]

here's how to do it without complements:
place the particular writer down in 1 position in the circle. then out of the 3 authors that aren't the ones that can sit next to the writer, there are 3x2 ways to place them adjacent to the writer, so u get something like AWA, 3 choices for the one on the left, 2 for the one on the right
then there are 4!x3! to place the remaining writers and authors, so in total, there are 3!3!4! = 864 ways to do this

with complements, we take AWA to be a single group, where the two A's are the authors that have to sit with the one particular writer W. There are then 8 total groups, where we need to place W's next to the AWA. So in total, theres 7x4!3!x2 = 2016 ways to put the artists with the one particular writer (the two is there because A1WA2 and A2WA1 and both valid configurations). then we get that there are 2880-2016 = 864 ways to do part b by the complement

this is one of the rare cases where complement is harder than just consider the case by itself