10. Find the number of positive integer solutions $(x, y, z, w)$ to the indeterminate equation $x+y+z+w=25$ that satisfy $x<y$.
Let $Y-X=\alpha, \alpha \in \mathbf{N}^{*}$, then $2 x+\alpha+z+w=25$, when $x$ takes values from $1 \sim 11$, the number of positive integer solutions for $\alpha+z+w=25-2 x$ is $C_{24-2}^{2}$, so the total is $C_{2}^{2}+C_{4}^{2}+C_{6}^{2}+\cdots+C_{22}^{2}=946, C_{2 n}^{2}=n(2 n-1)$, thus
$$
\sum C_{2 n}^{2}=2 \sum n^{2}-\sum n=\frac{n(n+1)(2 n+1)}{3}-\frac{n(n+1)}{2}=\frac{n(n+1)(4 n-1)}{6}
$$