Learn English – Choose the right variant

Both are valid, but they say very slightly different things. Carlo mentioned you can say "I chose from the menu" but note that you cannot say "*I chose among the menu" nor "*I chose from among the menu." I think that gives us the key to the difference.

When you say "chose [from] among" I assume you mean six physical ties. Now that you've taken one, only five are left. But when you say "chose from" I entertain the possibility that you mean six kinds of tie. That is, maybe you were shopping on Amazon, and there are still six ties for the next customer to consider.

This is a rule that runs at large. The difference between two expressions often cannot be seen without some context around them.

Very strictly speaking, variation is change, and a variant is one of the forms resulting from the change.

The use of variation to mean variant is so common, though, that only a hardcore pedant would ever even recognize a difference in that context, much less say either one is incorrect.