\[\boxed{\mathbf{178.}}\]
\[\textbf{а)}\ C_{20}^{10} = \frac{20!}{10!10!} =\]
\[= 11 \cdot 13 \cdot 2 \cdot 17 \cdot 19 \cdot 2 =\]
\[= 143 \cdot 34 \cdot 38 = 184\ 756.\]
\[\textbf{б)}\ 1.\]
\[\textbf{в)}\ \frac{1}{184756}.\]