Question

How do you find out how many ways the integers 1,2,3,4,5,6,7,8 can be placed in a circle if: at least three odd numbers are together?

Answer 1

360

Explanation:

Let's first look at the number of ways we can place three odd numbers together. That's a permutation question:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`

`P_(4,3)=(4!)/((4-3)!)=(4!)/(1!)=24`

The rest of the numbers (5) can be placed in any order:

`P_(5,5)=(5!)/((5-5)!)=5! =120`

The one other thing we need to remember is that this is a "sitting in a circle" problem - and so unlike sitting in a row (where there is definitely a chair 1 with person A in it, chair 2 with person B, etc), we have to eliminate the duplicate arrangements from simply rotating the table chairs. Since there are 8 people, there are therefore 8 ways to do each arrangement of chairs, and so we divide by 8.

In total then, we have:

`(24xx120)/8=360`