It is said that Prophet Muhammad (Sallallahu Alaihi wa Sallam) prayed only three Taraweeh prayers in the mosque. From fourth night onwards he prayed at home. If this is so then from where did he pray Taraweeh in the last 10 days of Ramadan, when Prophet (Sallallahu Alaihi wa Sallam) was performing I'tikaaf?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions. For Taraweeh, please read Fatwa: 82507. For your second question, it was narrated on the authority of Ayisha (may Allah be pleased with her) that the Prophet (peace be upon him) made I'tikaf during the last ten days of the month of Ramadan . This narration is in Al-Bukhari and Muslim and in other references. The point of difference among the scholars is the time when I'tikaf should begin. The majority of scholars including Abu Hanifa , Malik , Ash-Shafi'e , and Ahmad see that it should begin before sunset in order to complete ten days and ten nights. Yet, other scholars see that it should begin with the beginning of the day. The preponderant opinion, however, is the first one. The reason behind this difference of opinions is what Imam Muslim (may Allah have mercy on him) narrated in his Sahih (authentic book of Hadith) that Ayisha (Radiya Allahu Anhu) said: "whenever the Prophet (Sallallahu Alaihi wa Sallam) intended I'tikaf he would perform the Fajr (dawn) prayer, then enter his niche (secluded place for worship)" . An-Nawawi (may Allah have mercy on him) said: The advocates of the second opinion based their assumption on this narration. Al-Awzai'e , Ath-Thawri , and Al-Laith (in one of his two opinions) were also advocates of this opinion. Imam Malik , Abu Hanifa , Ash-Shafi'e , and Ahmad say that if one is to enter the state of I'tikaf for a month or ten days, he has to begin before sunset. They took the afore-mentioned Hadith of Ayisha to mean that the Prophet (Sallallahu Alaihi wa Sallam) entered his place of seclusion and stayed in isolation for worship after the Fajr prayer, not as the beginning of his I'tikaf. Rather he was already in the state of I'tikaf before sunset, but upon praying Fajr, he returned to his seclusion. Allah knows best.