An official count of people living in a place

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intimacy, passion and commitment are the components of a interpersonal relationship emphasized in the triangular theory of love by robert sternberg.

the triangular theory of love says love can be comprehend as this componest form the vertices of a triangle.  

intimacy relates to feelings such as of closeness and connection;

passion relates the drives that brings people together, as physical attraction.

commitment relates to the being in a long-term relationship.

d. i assume the underlined word is "role." reformat if it isn't.

explanation:

role is closest to part because when replacing words, "playing a vital decision," "playing a vital activity," and "playing a vital dollar" don't make logical sense. the only meaningful choice is "playing a vital part."

this fits one of merriam-webster's definitions of role: "a function or part performed especially in a particular operation or process."