10. For positive integer $n$ and real number $x (0 \leqslant x < n)$, define
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. If integers $m, n \geqslant 2$ satisfy
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$

find the value of $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$.
$$
\begin{array}{l}
f\left(m, \frac{1}{n}\right)=\left(1-\frac{1}{n}\right) C_{m}^{0}+\frac{1}{n} C_{m}^{1}, f\left(m, \frac{2}{n}\right)=\left(1-\frac{2}{n}\right) C_{m}^{0}+\frac{2}{n} C_{m}^{1}, \cdots, \\
f\left(m, \frac{n-1}{n}\right)=\frac{1}{n} C_{m}^{0}+\left(1-\frac{1}{n}\right) C_{m}^{1}, f(m, 1)=C_{m}^{1}, \\
\text{Thus } f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right) \\
=\frac{n-1}{2}\left(C_{m}^{0}+C_{m}^{1}\right)+C_{m}^{1}+\frac{n-1}{2}\left(C_{m}^{1}+C_{m}^{2}\right)+C_{m}^{2}+\cdots+\frac{n-1}{2}\left(C_{m}^{m-1}+C_{m}^{m}\right) \\
=\frac{n-1}{2}\left(C_{m}^{0}+C_{m}^{m}\right)+n\left(C_{m}^{1}+C_{m}^{2}+\cdots+C_{m}^{m-1}\right)=n-1+n \cdot\left(2^{m}-2\right) \\
=n \cdot 2^{m}-n-1=123 \Rightarrow n\left(2^{m}-1\right)=124=4 \cdot 31, \text{ then } n=4, m=5 . \\
\text{Therefore } f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right) \\
=m \cdot 2^{n}-m-1=5 \cdot 2^{4}-5-1=74 .
\end{array}
$$