What is the maximum number of subsets, having property that none of them is a subset of another, can a set with 10 elements have?

$\textbf{(A)}\ 126  \qquad\textbf{(B)}\ 210  \qquad\textbf{(C)}\ 252  \qquad\textbf{(D)}\ 420  \qquad\textbf{(E)}\ 1024$
1. **Understanding the Problem:**
   We need to find the maximum number of subsets of a set with 10 elements such that none of these subsets is a subset of another. This is a classic problem that can be solved using Sperner's theorem.

2. **Applying Sperner's Theorem:**
   Sperner's theorem states that the largest family of subsets of a set with \( n \) elements, where no one subset is a subset of another, is given by the binomial coefficient \(\binom{n}{\lfloor n/2 \rfloor}\).

3. **Calculating the Binomial Coefficient:**
   For a set with 10 elements (\( n = 10 \)):
   \[
   \left\lfloor \frac{10}{2} \right\rfloor = 5
   \]
   Therefore, we need to calculate \(\binom{10}{5}\):
   \[
   \binom{10}{5} = \frac{10!}{5! \cdot 5!}
   \]
   Breaking it down:
   \[
   10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
   \]
   \[
   5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
   \]
   Thus,
   \[
   \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = \frac{30240}{120} = 252
   \]

4. **Conclusion:**
   According to Sperner's theorem, the maximum number of subsets of a set with 10 elements, where no subset is a subset of another, is 252.

The final answer is \(\boxed{252}\)