They put an internal timer on them. If you wait about 5 minutes after killing the elite to use your 3oC your chances should improve.
I normally just run the highest lvl strikes and pop a coin at or right before I kill the end boss. Generally get one every 4-5 coins
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5 minutes before or after killing the boss
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Yep, which is how long it usually takes from strike end boss to the next one if your farming
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If only Bungie was as transparent as you good sir.
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No worries, good luck farming. Remember to be at your highest light lvl too when turning them in. If your below 300LL I have never seen them yield a 300 or higher but you have a higher chance of pulling a 310 if your above 300LL