Let's say x is the number of administrators, so the number of faculty members is 2x.

Then, we can careful count all the bows that took place:

- Every administrator (x) bows to each other administrator (x-1): total x(x-1)
- Every administrator (x) bows to each faculty member (2x): total x(2x)
- Every administrator (x) bows to Sr. Janet (1): total x
- Every faculty member (2x) bows to each other faculty member (2x-1): total 2x(2x-1)
- Every faculty member (2x) bows to each administrator (x): total 2x(x)
- Every faculty member (2x) bows to Sr. Janet (1): total 2x

^{2}-x+2x

^{2}+x+4x

^{2}-2x+2x

^{2}+2x=9x

^{2}

Therefore, 900=9x

^{2}, so x=10.

This means there were 10 administrators, 20 faculty, and Sr. Janet in attendance, for a total of 31 people.

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