\[C_{10}^{5} = \frac{10!}{5!(10 - 5)!} =\]
\[= \frac{10 \bullet 9 \bullet 8 \bullet 7 \bullet 6 \bullet 5!}{5 \bullet 4 \bullet 3 \bullet 2 \bullet 5!} =\]
\[= 3 \bullet 2 \bullet 7 \bullet 6 = 252\ комиссии.\]
\[Ответ:\ \ 252.\]