Is it obligatory to adhere to a specific order for the remembrances (adhkar) after prayer, or is a Muslim permitted to recite them however they wish?

All the prescribed remembrances (adhkar) after prayer are authentically reported from the Prophet, peace and blessings be upon him. The order mentioned is permissible, not obligatory. It is recommended to begin with seeking forgiveness three times, and then saying: "O Allah, You are As-Salam (the Source of Peace), and from You is all peace; blessed are You, O Possessor of majesty and honor."

It is also recommended to remember Allah and supplicate after the salutation. Among the reported remembrances are: "There is no god but Allah, alone, without partner. To Him belongs all sovereignty and all praise, and He is over all things competent. O Allah, there is none to withhold what You give, and none to give what You withhold, and the might of the mighty does not benefit them against You." And glorifying Allah (Tasbih), praising Allah (Tahmid), and magnifying Allah (Takbir) thirty-three times each, and saying: "There is no god but Allah, alone, without partner. To Him belongs all sovereignty and all praise, and He is over all things competent. There is no might or power except with Allah, the Most High, the Most Great. There is no god but Allah, and we worship none but Him. To Him belong all blessings, grace, and excellent beautiful praise. There is no god but Allah, sincere in devotion to Him, even if the disbelievers detest it." And seeking refuge with: "O Allah, I seek refuge in You from cowardice, and I seek refuge in You from miserliness, and I seek refuge in You from the feebleness of old age, and I seek refuge in You from the trials of this world and the torment of the grave." Raising one's voice in remembrance when people departed from the prescribed prayer was practiced during the time of the Prophet, peace and blessings be upon him.