Concept:
India has several renowned institutions dedicated to the promotion, preservation, teaching, and research of Indian classical dance and music traditions. These institutions have contributed significantly to performing arts education and cultural development.
Step 1: Identify the Indian dance institutions.
\[
\text{Kalamandalam}
\]
is a famous institution of Kerala established for the promotion of traditional performing arts such as Kathakali and Mohini Attam.
\[
\text{Nalanda Dance Research Centre}
\]
is an important institution dedicated to dance education and research.
\[
\text{Kalakshetra}
\]
is a renowned institution founded by Rukmini Devi Arundale for Bharatanatyam and Indian arts education.
\[
\text{Gandharva Mahavidyalaya}
\]
is a major institution for Indian classical music and dance education.
Thus:
\[
A,\ B,\ C,\ D
\]
are Indian performing arts institutions.
Step 2: Identify the institution that is not an Indian dance institution.
\[
\text{Trinity School of Music}
\]
is not an Indian dance institution.
It is associated with:
\[
\text{Western Music Education}
\]
Thus:
\[
E
\]
is excluded.
Step 3: Select the correct combination.
The correct set is:
\[
A,\ B,\ C,\ D
\]
Step 4: Compare with the given options.
Among the given choices:
\[
(A)\ A,E,C,D
\]
\[
(B)\ B,C,E,D
\]
\[
(C)\ A,E,D,B
\]
\[
(D)\ A,B,C,D
\]
the correct answer is:
\[
\boxed{A,B,C,D}
\]
Hence:
\[
\boxed{\mathrm{(D)}}
\]