You cannot open both trading and demat accounts entirely in a partnership firm's or private trust's own name.
According to CDSL operating guidelines (PDF), while you can open the trading account in the partnership firm's or private trust's name, the demat account is opened in the names of authorised individuals. For partnership firms, the demat account is opened in the names of a minimum of two authorised partners. For private trusts, the demat account is opened in the names of authorised trustees.
Additionally, both the trading and demat accounts are registered with the partnership firm's or trust's PAN number rather than the individual partners' or trustees' PAN numbers.