There were 50 faculty members comprising 30 males and the rest females. No male faculty member knew music, but many of the female faculty members did. The head of the institution invited six faculty members to a tea party by draw of lots. At the party it was discovered that no member knew music. The conclusion is that :

The correct option is "nothing can be said about the gender composition of the party."

Let us analyze the given information step-by-step to understand why this conclusion is the only logically sound one:

1. Faculty Composition:
There are 50 faculty members in total. Of these, 30 are males. Since the rest are females, we have:
50 - 30 = 20 female faculty members.

2. Knowledge of Music:
- We are told that no male faculty member knew music. Thus, all 30 males do not know music.
- We are told that many of the female faculty members did [know music]. This implies that some female faculty members know music, while other female faculty members do not know music.

3. The Tea Party Selection:
The head of the institution selected 6 faculty members by a draw of lots (random selection). At the party, it was observed that no member knew music. This means that all 6 attendees do not know music.

4. Analyzing the Attendees' Genders:
Who among the faculty members does not know music?
- All 30 males do not know music.
- The subset of females who do not know music.
Since the only condition for the attendees is that they do not know music, the 6 selected members could be:
- Entirely males (since males do not know music).
- Entirely females who do not know music.
- A mix of males and females who do not know music.

Because any of these gender combinations are possible and satisfy the condition that no one at the party knew music, we cannot determine the exact gender composition of the party. Therefore, nothing can be said about the gender composition of the party.