It is not permissible
to vow to fast the whole month of Dhul-Hijjah. This month includes days during which fasting is prohibited; these are the Days of Tashriq (11th, 12th and 13th of Dhul-Hijjah) and `Eid-ul-Adha (the Festival of the Sacrifice). Since your mother failed to fast Dhul-Hijjah supposing that she excluded the days during which it is unlawful to fast, she has to make a Kaffarah (expiation) for a broken oath because she could not fulfill her vow.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.