Select an acceptable pronoun to complete this sentence.
every contestant should wear her or histheirititshe or shethey best suit on tuesday.

The correct pronoun is their.

Every contestant should wear their best suit on Tuesday.

Their is a personal pronoun that means belonging to or associated with the people or things previously mentioned or easily identified. Therefore the suits belongs to the contestants, there is more than one contestant, we don't know if they are male or female and 'it' refers to objects not people so the correct pronoun is their.

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