Q. When a smooth card placed over the mouth of a tumbler is flicked sharply in the horizontal direction , the card flies away , but the coin kept over the card falls into the tumbler.

Yes it happens due to  inertia of rest of coin. When the card is flicked the coin tends to remain in its state of rest while the card  moves away and thus it falls into tumbler
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Before we flicking the card, the card as well as the coin were at rest. But when we flicked the card, the card started moving in a motion. Since the coin is at rest, it will fall down by the first law of motion.
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Both the card and the coin are at rest in the beginning. When we flick the card, the coin on top of it tends to remain in rest due to inertia of rest, and hence it stays in the same place and falls into the tumbler.
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It is because of the inertia of rest. When a card is flicked horizontally the card gets into motion but the coin (due to the inertia of rest) remains into rest.
Regards
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