पृथग्विनानानाभिस्तृतीयाऽन्यतरस्याम्

Operational rule vidhi

When joined with the words पृथक् 'without', विना 'without' and नाना 'without', the 3rd case-affix is used, optionally (as well as the 5th and the 2nd).

Word by word (padaccheda)

With / by पृथक्- विना- नानाभिः tṛtīyā · bahuvacanaSubstitute / term तृतीया prathamā · ekavacanaRight context अन्यतरस्याम् saptamī · ekavacana

With / by Tṛtīyā (3rd case): accompaniment or means.
Substitute / term Prathamā (1st case): in operational rules, the substitute or the thing introduced; in definitions, the term or its referent.
Right context Saptamī (7th case): "when X follows", the operation applies to what precedes (1.1.66 तस्मिन्निति निर्दिष्टे पूर्वस्य). Also a condition or domain ("in the sense of").

Full reading

पृथक्-विना-नानाभिः तृतीया अन्यतरस्याम्

Pāṇini leaves out words that carry over from earlier sūtras. Expand to see the rule as a reader actually understands it.

Commentary

When joined with the words पृथक् 'without', विना 'without' and नाना 'without', the 3rd case-affix is used, optionally (as well as the 5th and the 2nd).
Ex. रामाद्रामेण रामं वा विना पृथग् नाना वा 'without or different from Rama.'

This sutra may be divided into two parts:- (1) पृथग् विना नानाभिः (2) तृतीयाऽन्यतरस्याम्. By this arrangement we can read dvitiya into the aphorism.

विना वातं विना वर्षं विद्युत्प्रपतनं विना ।
विना हस्ति कृतान्दोषान् केनेमौ पातितौ द्रुमौ ॥

Śrīśa Chandra Vasu, The Ashtadhyayi of Panini (1897), via ashtadhyayi.com.