Let $O$ be the set of odd numbers between 0 and 100.  Let $T$ be the set of subsets of $O$ of size $25$.  For any finite subset of integers $S$, let $P(S)$ be the product of the elements of $S$.  Define $n=\textstyle{\sum_{S \in T}} P(S)$.  If you divide $n$ by 17, what is the remainder?
1. **Identify the set \( O \) and the set \( T \):**
   - \( O \) is the set of odd numbers between 0 and 100, i.e., \( O = \{1, 3, 5, \ldots, 99\} \).
   - \( T \) is the set of subsets of \( O \) of size 25.

2. **Define \( n \) as the sum of the products of elements of each subset \( S \in T \):**
   - \( n = \sum_{S \in T} P(S) \).

3. **Use Vieta's formulas to relate the problem to polynomial coefficients:**
   - We need the sum of the products of 25 distinct elements from \( O \). This is equivalent to finding the coefficient of \( x^{25} \) in the polynomial \( (x-1)(x-3)\ldots(x-99) \).

4. **Simplify the polynomial modulo 17:**
   - Since \( x \equiv x \pmod{17} \), we can reduce the polynomial modulo 17.
   - The polynomial \( (x-1)(x-3)\ldots(x-99) \) modulo 17 can be written as:
     \[
     (x-1)(x-3)\ldots(x-15)(x-19)\ldots(x-33)(x-37)\ldots(x-51)(x-55)\ldots(x-69)(x-73)\ldots(x-87)(x-91)\ldots(x-99).
     \]
   - Grouping terms and reducing modulo 17, we get:
     \[
     (x-1)(x-2)\ldots(x-16).
     \]

5. **Express the polynomial in terms of \( x^{16} \) using Fermat's Little Theorem:**
   - By Fermat's Little Theorem, \( x^{16} \equiv x \pmod{17} \).
   - Therefore, the polynomial can be written as:
     \[
     (x-1)(x-2)\ldots(x-16) \equiv (x^{16}-1) \pmod{17}.
     \]

6. **Find the coefficient of \( x^{25} \) in the polynomial:**
   - We need the coefficient of \( x^{25} \) in \( (x^{16}-1)^3 \).
   - Simplify \( (x^{16}-1)^3 \):
     \[
     (x^{16}-1)^3 = x^{48} - 3x^{32} + 3x^{16} - 1.
     \]
   - We need the coefficient of \( x^{22} \) in \( \frac{(x^{16}-1)^3}{x+1} \):
     \[
     \frac{(x^{16}-1)^3}{x+1} = (x^{32}-2x^{16}+1)(x-1)(x^{14}+x^{12}+\cdots+1).
     \]

7. **Calculate the coefficient of \( x^{22} \):**
   - The coefficient of \( x^{22} \) in \( (x^{32}-2x^{16}+1)(x-1)(x^{14}+x^{12}+\cdots+1) \) is \( -2 \) times the coefficient of \( x^6 \) in \( (x-1)(x^{14}+x^{12}+\cdots+1) \).
   - The coefficient of \( x^6 \) in \( (x-1)(x^{14}+x^{12}+\cdots+1) \) is \( -1 \).

8. **Determine the final coefficient and its negative modulo 17:**
   - The desired coefficient is \( 2 \).
   - Its negative is \( -2 \equiv 15 \pmod{17} \).

The final answer is \( \boxed{15} \).