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(C) Let's analyze each pair:$(A)$ $CH_2=CH-OCH_3$ (methoxyethene) and oxetane $(C_3H_6O)$ are ring-chain isomers because one is an open-chain compound

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(C) Let's analyze each pair:$(A)$ $CH_2=CH-OCH_3$ (methoxyethene) and oxetane $(C_3H_6O)$ are ring-chain isomers because one is an open-chain compound and the other is a cyclic compound with the same molecular formula.$(B)$ $o$-methylbenzyl alcohol $(C_8H_{10}O)$ and $2,3$-dimethylphenol $(C_8H_{10}O)$ are functional isomers because they contain different functional groups (alcohol vs. phenol).$(C)$ Butan-$2$-ol and butan-$1$-ol are position isomers because the position of the $-OH$ group changes,not the carbon chain length. Therefore,calling them chain isomers is incorrect.$(D)$ Pent-$1$-ene and $2$-methylbut-$1$-ene are chain isomers because the parent carbon chain length changes from $5$ to $4$ carbons.

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