So this problem was used for a while in interviews - I thin think that may be why you see the "broken" version of it, like the one I gave, so often.
What you need to fix the problem is the constraint that only one person can cross the bridge at a time. With the "broken" problem you send the slowest person across on the bicycle, while the other two people cross and everyone is across the bridge in 5 minutes.
If only one person can cross the bridge at a tiem you send the slowest person across on the bike, then the fastest person crosses the bridge, then comes back on the bike, and sends the next slowest person back on the bike before crossing themselfe. So we have:
10 min person crosses on the bike, time 1 min
2 min person crosses, time 2 mins
2 min person comes back on bike, time 1 min
5 min person crosses on bike, time 1 min
2 min person crosses, time 2 mins
The total crossing time is 1+2+1+1+2 or 7 minutes.
The new twist that the Mathproblems.info guys put on the problem was to remove the restriction that only one person could cross at a time, and allowing the bike to be left anywhere on the bridge. I am still working on this new version of the problem.