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Ternary (or trinary) stars are not the same as triple stars. So, what is the basic underlying difference between these two? What are some popular examples of these two varieties?

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Exactly the difference between binary and double stars.

Trinary System

Three stars gravitationally bound to each other.

Here are some examples

Triple Star System

Three stars that appear to be together but any one of them is gravitationally unbound.

Here are some examples.

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