Final answer:
The word 'there' correctly completes the sentence as it refers to a place or position, whereas 'their' indicates possession and 'they're' is a contraction for 'they are.'
Step-by-step explanation:
The correct answer to your question about which word completes the sentence 'How many books are on the shelf where they left them?' is option 1) there. The word 'there' refers to a place or position, which is what is being asked in the context of this question. The other options, 'their' and 'they're,' are incorrect in this context because 'their' is a possessive pronoun that shows ownership, and 'they're' is a contraction of 'they are' which does not fit the sentence structurally or meaningfully.