No, but the number of council tax paying adults you live with does affect the level of reduction you can receive.
If you are diagnosed as SMI by a doctor, entitled to at least one of the applicable benefits for the discount, and living alone or only with others who are SMI, you are exempt from paying council tax – you won’t have to pay anything at all.
Alternatively, if you are diagnosed as SMI by a doctor, entitled to at least one of the applicable benefits for the discount, but live with one adult who is eligible to pay council tax, your household will receive a 25% reduction. This is because as you are exempt from paying council tax, the other inhabitant will qualify for the single person’s council tax discount.
However, even if you have been diagnosed as SMI by a doctor and are entitled to at least one of the applicable benefits for the discount, but you live with two or more adults who are eligible to pay council tax, there will be no reduction. As there are two or more full council tax paying adults, they are not eligible for the single person’s council tax discount.