Okay, I'll revise my count and way of thinking:
Rather than considering things per-hour, I'll consider the number of times a hand "laps" another hand, that is to say the number of times it does 360* more than the other hand. For that to happen, the faster hand must have been in right angles twice. We know which hand is faster, so recompute:
Hour-Minute: 24 hours = 2 loops (hour hand), 24 hours = 24 loops (minute-hand), thus the minute hand passes the hour hand 22 times => 44 loops.
Hour-Second: 24 hours = 1440 loops (second-hand), thus the second hand passes the hour hand 1440 - 2 = 1438 times => 2876 loops.
Minute-Second: 24 loops (minute-hand) vs. 1440 loops (second-hand), thus we have 1416 passes => 2832 loops.
The total is 5752 loops.
We have 3 hands and therefore 3 "pairs" to consider. They are the hour-minute, hour-second, and minute-second.
Each hour, the minute hand will be at a right angle twice, so that's 48 right angles -- 2 for each hour of the day.
Each hour, the second hand will go around 60 times and have 2 right angles for each, that's 120 right angles per hour or 120 * 24 = 2880.
Each minute, the second hand will make 2 right angles with the hour hand, that's another 120 * 24 = 2880.
The total is 5808.