Question
Did Lord Rama reach Ayodhya after 20 days from Dusherra?
Answer
I do not know how the story of Rama killing Ravana on Vijaya Dashami and returning on Deepawali became famous.
If we go through Valmiki Ramayana. It doesn't give precise information on time span between killing of Ravana and returning of Rama to Ayodhya. Furthermore, it is not possible that Rama killed Ravana on Vijaya Dashami and awaited 20 days there and returned to Ayodhya on Deepawali, as they returned through Pushpak Vimana.
Padma Purana Patala Khanda chapter 36 gives each and every Tithi of Rama's life in detail. The relevant verses from that Chapter are:
संस्कारो रावणादीनाममावस्यादिनेऽभवत् ॥
वैशाखादितिथौ राम उवास रणभूमिषु ।
अभिषिक्तो द्वितीयायां लङ्काराज्ये विभीषणः ॥
सीताशुद्धिस्तृतीयायां देवेभ्यो वरलम्भनम् ।
हत्वा चिरेण लङ्केशं लक्ष्मणाग्रज एव सः ॥
गृहीत्वा जानकीं पुण्यां दुःखितां राक्षसेन तु ।
आदाय परया प्रीत्या जानकीं स न्यवर्तत ॥
वैशाखस्य चतुर्थ्यां तु रामः पुष्पकमाश्रितः ।
विहायसा निवृत्तस्तु भूयोऽयोध्यां पुरीं प्रति ॥
पूर्णे चतुर्दशे वर्षे पञ्चम्यां माधवस्यतु ।
भरद्वाजाश्रमे रामः सगणः समुपाविशत् ॥
नन्दिग्रामे तु षष्ठ्यां स भरतेन समागतः ।
सप्तम्यामाभिषिक्तोऽसावयोध्यायां रघुद्वहः ॥
दशैकाधिकमासांस्तु चतुर्दशाहानि मैथिली ।
उवास रामरहिता रावणस्य निवेशने ॥
द्विचत्वारिंशक वर्षे रामो राज्यमकारयत् ।
सीतायाश्च त्रयस्त्रिंशद्वत्सराश्च तदाऽभवन् ॥
स चतुर्दशवर्षान्ते प्रविश्य च पुरीं प्रभुः।
अयोध्यां मुदितो रामो हत्वा रावणमाहवे॥
भ्रातृभिः सहितस्तत्र रामो राज्यमथाकरोत्।
The obsequial ceremonies of Ravana and others took place on the new-moon day. Rama stayed on the battlefield on the first day of Vaisakha. On the second day of Vaisakha Vibhisana was consecrated on the kingdom (i.e. as the king) of Lanka. On the third day took place the purification of Sita and obtaining boons from gods. Having, after a long time, killed the lord of Lanka, he, the elder brother of Laksmana, accepted the auspicious Janaki, afflicted by the demon (viz. Ravana) and taking her (with him) with great love, he returned. On the fourth day of Vaisakha Rama got into the Puspaka and through the sky again came back to Ayodhya. When the fourteenth year was complete Rama along with his group stayed in Bharadvaja's hermitage on the fifth day of Vaisakha. On the sixth day he met Bharata at Nandigrama. On the seventh the descendant of Raghu (i.e. Rama) was consecrated (as the king) in Ayodhya. MaithilI (i.e. SIta) lived without Rama in the house of Ravana for eleven months and fourteen days. Rama ruled (i.e. was consecrated as the king) when he was forty-two years old. At that time SIta had completed thirty-three years. That lord Rama being delighted entered the city Ayodhya at the end of the fourteenth year and with his brothers ruled there.
Thus Rama got into Pushpaka on 4th of Vaisakha. He lived in Bhardwaja Ashrama on 5th day of Vaisakha. He met Bharata at Nandigrama on 6th day of Vaisakha. Rama was Consecrated as king on 7th day of Vaisakha.
As Deepawali and VinayaDashami falls on Ashwin Kartik and considering 20 days gap between them, it is not possible that Rama killed Ravana on Dashami and returned on Deepawali.
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