Hello, saintv!

It depends on the interpretation of "the 3 boys are not together".How many ways can 8 boys be arranged in a circle

if 3 boys cannot sit next to each other?

This is how I solved it:

(Total ways boys can be arranged) - (3 boys together) .= .(Ways 3 boys are not together)

So, I did: . ways

I think this is wrong though.

Suppose the eight boys are: {A, B, C, D, E, X, Y, Z}

. . and X, Y, Z must not sit in adjacent chairs.

If "the 3 boys are together" means (in some order),

. . your solution is correct.

Note that this allows, for example: . X A Y Z B C D E

. . whereof them can be together.two

Ifnoof those boys can be adjacent, a different approach is used.two

Place the other 5 in a circle; there are: . ways.

. . There are 5 spaces between these 5 boys.

Select 3 of these spaces to place X, Y, Z.

. . There are: . ways.

Therefore, there are: . ways.