How did the Prophet allow a woman to sing with the drum, knowing that a woman's voice is 'awrah (something that must be concealed)?

A woman's voice is not considered 'awrah' (private and unlawful to expose) in an absolute sense. Rather, it is permissible to hear it as long as there is no submissiveness in speech that could lead to temptation (fitna). Evidence for this includes the allegiance (bay'ah) of women to the Prophet, peace and blessings be upon him, and the singing of a young girl in front of the Prophet, peace and blessings be upon him. The four madhhabs (schools of thought) are in agreement on this matter.