Is it permissible for the mute person to utter the Tasmiyah (saying Bismillah) if they are able to do so?

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It is permissible for the mute person to utter the Tasmiyah and they are not prohibited from doing so. Prohibition applies only regarding the Quran, not general remembrance (Dhikr). Therefore, it is prescribed for a person in a state of major ritual impurity (Janabah) to say Bismillah when bathing, and their state is less severe than that of a disbeliever who is permitted to say Bismillah when slaughtering. Al-Hasan, Al-Hakam, Al-Laith, Al-Shafi'i, Ishaq, Abu Thawr, and the companions of opinion permitted the slaughter of one in a state of Janabah.