Step 1: Recall nucleosome structure.
A nucleosome core consists of histone proteins around which DNA is wrapped.
Step 2: Histone composition.
The histone octamer contains:
\[
2 \times (\text{H2A, H2B, H3, H4}) = 8~\text{histone polypeptide chains}
\]
Step 3: Conclusion.
\[
\boxed{\text{Number of polypeptide chains} = 8}
\]