in the position to

they are NOT the same
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(1) They were in the position to carry out the mission

Bad English. Discard. "The position"? What position is that?

"In THE position to..." is not a normal English usage.
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(2) they were in a position to carry out the mission

"In A position to..."

The "position" is metaphorical, not physical or geographic.

This means they were able to carry out the mission.

I am now in A position to pay my debts (I now have some money)

I am in A position to go outside (I have my shoes on)

I am in A position to marry you. (I am not already married, and I am over 16)
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(3) they were in position to carry out the mission

"In position to..."

here the "position" is physical / geographic

They were in the right places to carry out the mission.

The police were in position around the drug factory. (The police had surrounded the drug factory)

I was in position in case the old man fell. (I was next to him)
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Literarypoland, it would be nice to get some acknowledgement sometimes! (It's good manners to say "thank you").

Thanks, contrex. You are one of those I can always count on.

I apologise for what I said. I am happy to help you always.