Thank you for the opportunity to help you with your question!
arrange 3 white bead on first, there is 1 way to do this (because the beads are identical). Then arrange 3 black in 3 spaces between the white beads, when 3 black beads are next to each other there is only one pattern (no matter whichever space they take), when no black beads next to each other there is also only one pattern, when 2 black bead next to each other and separate with the third black bead, there are two patterns 1 black bead on the left or on the right of the two black beads, so total # of patterns is 4.
Please let me know if you need any clarification. I'm always happy to answer your questions.