Quick Permutations Question (1 Viewer)

[pikachu975]

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

It's from the Fitzpatrick book. I got the first part by doing 1 * 4! * 5! = 2880 but I can't do the second part. I was thinking 2880 - the number of arrangements with 2 artists sitting next to the write but I can't figure it out. Any help appreciated.

 

[Drsoccerball]

Try 'tying' the 3 people together noticing that only the two can be arranged among the 3. The rest is simple.

 

[pikachu975]

Try 'tying' the 3 people together noticing that only the two can be arranged among the 3. The rest is simple.

I just don't get how to do it in a circular arrangement. If it was a line I could do it though. This is so confusing...

 

[si2136]

Find the arrangements of two sitting together, minus the original perm.

 

[trecex1]

I just don't get how to do it in a circular arrangement. If it was a line I could do it though. This is so confusing...

If they are either side of the particular writer - put those 3 as a group, you have 8 groups in total. 2! ways to arrange the two artists. In a circle = (8-1)! x 2!
And minus that from 9!

 

[pikachu975]

If they are either side of the particular writer - put those 3 as a group, you have 8 groups in total. 2! ways to arrange the two artists. In a circle = (8-1)! x 2!
And minus that from 2880

But 7! * 2! is more than 2880

 

[trecex1]

But 7! * 2! is more than 2880

Sorry minus that from 9! (total ways)

 

[leehuan]

One thing I'm assuming: There are specifically 2 artists that must not be beside the writer. If we had to select the two artists, we'd have to multiply by 5C2. But I don't think that's necessary...

 

[pikachu975]

Find the arrangements of two sitting together, minus the original perm.

One thing I'm assuming: There are specifically 2 artists that must not be beside the writer. If we had to select the two artists, we'd have to multiply by 5C2. But I don't think that's necessary...

The answer to part 1 is 2880 which I get is 4! * 5! but part 2 it says the answer is 864.

 

[InteGrand]

The second part of the question intends us to also meet the condition of the first part, i.e. the writers should still all be separated.

 

[pikachu975]

The second part of the question intends us to also meet the condition of the first part, i.e. the writers should still all be separated.

How would you do it considering this?

 

[trecex1]

How would you do it considering this?

Ok this is what i got: set the one particular writer then, 3P2 (rearranging the 2 artists in the 3 seats not next to the writer) x 3P3 (rearranging remaining 3 artists in the remaining 3 seats) x 4! (rearranging the remaining 4 writers) = 864

edit: these possible seats are from drawing out the alternating combination, for which there is only one way (as it is in a circle)

 

[pikachu975]

Ok this is what i got: set the one particular writer then, 3P2 (rearranging the 2 artists in the 3 seats not next to the writer) x 3P3 (rearranging remaining 3 artists in the remaining 3 seats) x 4! (rearranging the remaining 4 writers) = 864

edit: these possible seats are from drawing out the alternating combination, for which there is only one way (as it is in a circle)

Thanks so much