"Change their positions" vs. "change their position" [duplicate]

Possible Duplicate:
“Those who qualify will be awarded a certificate” or “those who qualify will be awarded certificates”?
“On their back” or “on their backs”?

Should the singular or plural of position should be used in the following sentences?

• Why Political Parties Change Their Positions: Environmental...
• Do the stars change their position through the course of the night?
• Why do stars appear to change their positions through the year?
• Why do shadows change their position during the day?

The sentences are from the first page of the Google search for should not change their position.

All of these examples should use the plural positions. Stars and parties are plural, obviously, but also note that each star or each party has a position — so that when plural the positions are plural.