Five methods of nomination used today include:

convention
caucus
direct primary
inheritance
promotion
petition
announcement

The five correct items are:

convention

caucus

direct primary

petition

announcement

Those are the means by which candidates enter the nomination process for public elections.   The two terms that don't fit are "inheritance" and "promotion."   An inheritance-based system would be like the old arrangements of monarchs or nobles, where someone took over the throne or provincial government simply because he was the son of the previous king or lord.  A promotion-based system would mean that persons advanced to positions of public office not through democratic election processes, but by being promoted internally within the government.  For some staff positions in government that does happen, and then those are appointed positions rather than elected positions.  But for elected positions in the government, democratic processes are followed.  Candidates for office must at least announce their candidacy.  They may need a certain number of petition signatures to get their names on the official election ballot.  Or they need to get on the ballot through preliminary steps by the larger political parties, such as caucuses, direct primaries, and the conventions of the parties.