Is it Sunnah to rinse the mouth and the nose together with the same handful of water?
All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is a Sunnah to rinse the mouth and nose from the same handful of water and this should be done three times. It is confirmed that the method of performing ablution by the Prophet is as narrated by 'Uthmaan who said: "…Then he put his hands (in the container of water) and pulled out water and rinsed his mouth and sniffed up water into his nostrils from the same handful of water, and he did this three times." [Muslim and Abu Daawood]
Besides, Al-Bukhari reported the Hadeeth with the following wording: "…Then he took a scoop of water and rinsed his mouth and sniffed up water into his nostrils."
Ibn Habbaan reported the Hadeeth with the following wording: "Three times with three handfuls."
Imaam An-Nawawi stated in his commentary on the book of Imaam Muslim : "There is a clear evidence in this Hadeeth about the correct method, and that the Sunnah in rinsing the mouth and sniffing up water into nostrils is that it should be done in three handfuls of water, rinsing the mouth and sniffing water into nostrils from each one of them." As regards raising the hands while making a supplication when entering the mosque and leaving it, then we do not know of any evidence about it in the Sunnah.
Allaah Knows best.
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