Is It Recommended to Fast 10 Days of Dhul-Hijjah?
Question

Could you please answer my questions regarding this month of Dhul-Hijjah? On your website, you mention how it is recommended to fast on the day of ‘Arafah, but I have read that this is true also of the first ten days of Dhul-Hijjah, is this correct? If it is, then can you confirm if in total there will be 9 not ten fasts as the tenth day is the Day of ‘Eid?
Praise be to Allah.
Is it recommended to fast 9 days of Dhul-Hijjah?
Fasting nine days of Dhul-Hijjah is recommended. This is indicated by the words of the Prophet (peace and blessings of Allah be upon him), as narrated in the hadeeth of Ibn 'Abbas (may Allah be pleased with him): 
“There are no days on which righteous deeds are more beloved to Allah than these ten days – i.e., ten days of Dhul-Hijjah).” They said: “O Messenger of Allah, not even jihad for the sake of Allah?” He said: “Not even jihad for the sake of Allah, unless a man goes out himself with his wealth and does not come back with anything (i.e., he expends all his wealth and is martyred).” (Narrated by al-Bukhari, 969)
And it was narrated from Hunaydah ibn Khalid from his wife that one of the wives of the Prophet (peace and blessings of Allah be upon him) said: “The Messenger of Allah (peace and blessings of Allah be upon him) used to fast nine days of Dhul-Hijjah and the day of ‘Ashura’ and three days of each month – the first Monday and two Thursdays.” (Narrated by Imam Ahmad, 21829; Abu Dawood, 2437; classed as da’eef in Nasb al-Raayah, 2/180, but classed as saheeh by al-Albani.) 
Is fasting on the day of `Eid permissible?
With regard to fasting on the day of ‘Eid , this is haram (prohibited), as is indicated by the marfoo’ hadeeth of Abu Sa’eed al-Khudri (may Allah be pleased with him): “He [the Prophet (peace and blessings of Allah be upon him)] forbade fasting on the day of ‘Eid) al-Fitr and the day of al-Nahr (the day of sacrifice, i.e., ‘Eid al-Adha).” (Narrated by al-Bukhari, no. 1992; Muslim, 827) 
The scholars are unanimously agreed that it is haram to fast on these days. 
Righteous deeds on these ten days  are better than on others. With regard to fasting, that should only be done on nine of them; the tenth day is the day of ‘Eid when it is haram to fast. 
Based on this, what is meant by the virtue of fasting the ten days of Dhul-Hijjah is fasting nine days only, even though they are called ten days. 
And Allah knows best.
Reference:
Sharh Muslim by al-Nawawi, hadeeth no. 1176.