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To calculate the expression \(106 \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10\), we can first find the factorial of 10, denoted as \(10!\), and then multiply it by 106.

The calculation of \(10!\) is:

\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800 \]

Now, we multiply this by 106:

\[ 106 \cdot 10! = 106 \cdot 3628800 = 384384000 \]

Thus, the final result is:

\[ 106 \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 = 384384000 \]