Changing Places

The above clock face indicates a little before 42 minutes past 4. 
The hands will again point at exactly the same spots a little after 23 minutes past 8. 
In fact, the hands will have changed places. 

How many times do the hands of a clock change places between three o'clock p.m. and midnight? 
And out of all the pairs of times indicated by these changes, what is the exact time when the minute hand will be nearest to the point IX?

See answer

Math Genius