Question 968110
Make a graph of the sequence, it will help to visualize

We need to keep 5 numbers:

Case 1: begin with 1 (
 case1A:  begin with 14 and choose 3 from (5789)----> 4 ways
 case1B1: begin with 12 and choose 3 from (5789)----> 4 ways
 case1B2: begin with 123 and then we left with (89)---> 1 way
 case1C:  begin with 15 and then we left with (789)----> 1 way

Case 2: Begin with 4 and then we left with (5789) -----> 1 way
Case 3: Begin with 2 and then we again left with (5789)----> 1 way

Hence total #ways = 12