Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa
Fasting the 9th and the 10th of Muharram is a Prophetic Sunnah. The Prophet said, “Fasting the day of Ashura, I hope that Allah will accept it as expiation for the previous year.” (Muslim) The Prophet also said: “If I live next year, I shall also fast on the 9th day.” (Ahmad)
Responding to the question, Sheikh M. S. Al-Munajjid, a prominent Saudi Muslim lecturer and author, states,
Abdullah ibn Abbas (may Allah be pleased with him) said: When the Messenger of Allah (peace be upon him) fasted on the day of Ashura and told the people to fast, they said, “O Messenger of Allah, this is a day that is venerated by the Jews and Christians.”
The Messenger of Allah (PBUH) said, “Next year, if Allah wills, we will fast on the ninth day.” But by the time the following year came, the Messenger of Allah had passed away. (Muslim)
Ash-Shafi`i and his companions, Ahmad, Ishaq, and others said: It is recommended to fast both the 9th and the 10th because the Prophet (PBUH) fasted the 10th and intended to fast the 9th.
Based on this, there are different ways of fasting Ashura, the least of which is to fast the 10th only, but it is better to fast the 9th as well. The more one fasts in Muharram, the better.
Allah Almighty knows best.
Source: www.islamqa.info