Final answer:
The correct pronoun to complete the sentence 'No one on this bus seems to know _____ way around this part of New York City' is 'their'. It's widely accepted to use 'their' as a singular pronoun when the gender is unknown or when referring to a generic individual.
Step-by-step explanation:
To complete the sentence with the correct pronoun, we should consider that 'no one' is an indefinite pronoun and is singular in form. Therefore, the pronoun that agrees with 'no one' should also be singular. The correct sentence is: 'No one on this bus seems to know their way around this part of New York City.'
In fact, using 'their' as a singular pronoun is increasingly accepted, especially when the gender of the antecedent (in this case, 'no one') is unknown or when it is used to refer to a generic individual, as in this sentence.