Question

4 boys and 2 girls sit in a line on stools in front of a coffe bar.
(a) in how many ways can they arrange themselves so that the two girls are together?
(b) in how many ways can they sit if the two girls are not together?

Answer 1

if the two girls are together, then there are 5 places they can be
G;s B B B B
B G;s B B B
B B G;s B B
B B B G;s B
B B B B G;s

each of these five has \(2\times 4!\) arrangements, the \(2\) for the two ways to arrange the two girls and the \(4!\) for the four boys

therefore your answer is
\[5\times 2\times 4!\]

Answer 2

if there were no restrictions at all, the answer is a clean \(6!\) so if the girls cannot sit together it would be \(6!-5\times 2\times 4!\)

Answer 3

thanks @satellite73

Answer 4

yw