Quick answerThe 16-year-old needs one more dose, the 13-year-old needs a third dose, and the 23-year-old needs 3 doses (options 4, 2, 6).
Why this is correctApply the rules to each record. The 16-year-old started at 14, before the 15th birthday, so 2 doses complete the series; dose 2 is now due. The 13-year-old received dose 2 only 4 months after dose 1, less than the 5-month minimum, so a third dose is needed. The 23-year-old is starting at 15 or older and is within routine catch-up through age 26, so 3 doses are needed.
Easy analogy or mental pictureThe HPV schedule is like a two-key lock opened by age at the first dose: start before 15 and 2 keys work, start later or with a weak immune system and you need 3 keys.
Memory hookStart under 15 = 2 doses at least 5 months apart; 15 or older, or immunocompromised = 3.
NCLEX decision ruleBase the dose count on the age at the first dose, the actual interval between doses, and immune status, not on current age alone.
Why the other choices are wrongThe 2-dose series for the 14-year-old on chemotherapy (option 5) is tempting because the child started before 15, but immunocompromised clients need 3 doses regardless of start age. Declaring the 19-year-old ineligible (option 3) is tempting because vaccine does not treat existing infection, but it still protects against HPV types not yet acquired, and catch-up is recommended through age 26. Restarting the lapsed series (option 1) is tempting because a long gap feels like lost protection, but vaccine series are resumed where they stopped and never restarted.
References
1CDC: Child and Adolescent Immunization Schedule by AgeHPV 2-dose and 3-dose schedules, minimum intervals, and immunocompromised dosing
2CDC: Adult Immunization Schedule by AgeHPV catch-up vaccination through age 26