Permutations Qs (1 Viewer)

[a1079atw]

Hi,
Can someone please explain the following question to me? I get the first part of the question but not the underlined second part...
"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?"
The answer is 4! x ³P₂ x 3!.

 

[Kurosaki]

Hi,
Can someone please explain the following question to me? I get the first part of the question but not the underlined second part...
"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?"
The answer is 4! x ³P₂ x 3!.

It's exactly as it says: two particular artists aren't allowed to sit next to a writer, for whatever reason - perhaps they insulted each other's artworks.
It is actually fairly simple:

Step 1: Arrange the 5 writers first. Simple, just 4!=24.
Step 2: You might like to draw a pentagon to visualise this. The vertices of the pentagon will represent the writers, the sides the artists. Circle any of the vertices of the pentagon, and let this particular point be the 'particular writer', which I'll call Fred. So, if the writers and artists must continue to be separated, then the logical conclusion is that the two artists, which we'll call A and B, cannot be on the sides with that vertex. There are then only 3 sides of the pentagon that A and B can occupy, since they despise Fred. Choose 2 of these sides - . Arrange them in 2! ways, and multiplying gives 6, which is .
Step 3: Order the rest of the artists - 3!=6 ways.
Multiply together to get 864.

 

[a1079atw]

Oh my god this explanation is genius Thank you, I totally get it now.