Your choice of "The character sharing his hopes for the future with his partner" as the correct answer is sensible because it does not pertain to a flashback or recollection of past events; rather, it focuses on the character's present or future aspirations.
Flashbacks typically involve revisiting past events or memories, so if a character is sharing their hopes for the future, it is not a reflection on their past, thus LEAST likely to be part of a flashback.
For example, if the character were to say, "I hope we can travel together one day," that statement is forward-looking and does not evoke any memories from the past.
On the other hand, the other options involve elements that are directly connected to past experiences:
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The character directly telling another person about this past event - This clearly involves recounting a memory and can function as a flashback.
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The character having a dream about this memory from the past - Dreams often pull from past experiences, making this statement also linked to a flashback.
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The character writing a letter reflecting on his past, revealing his thoughts to the reader - This directly implies a reflection on past experiences, clearly indicative of a flashback.
In summary, discussing the future does not relate to the concept of a flashback about childhood memories, making that option the least likely to appear in the context of a flashback.