10. Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be a permutation of the numbers $1,2,3,4,5$. If there do not exist $1 \leqslant i<j<k \leqslant 5$ such that $a_{i}<a_{j}<a_{k}$, then the number of all such permutations is $\qquad$.
According to the problem, the permutation $a_{1} a_{2} a_{3} a_{4} a_{5}$ does not contain any 3 numbers in ascending order.
For the permutations of $1,2,3,4$, there are 24 cases in total:
\begin{tabular}{llllllllllll}
1234 & 1243 & 1324 & 1342 & 1423 & 1432 & 2134 & 2143 & 2344 & 2341 & 2413 & 2431 \\
3124 & 3142 & 3214 & 3241 & 3412 & 3421 & 4123 & 4132 & 4213 & 4231 & 4312 & 4321
\end{tabular}

There are 14 permutations that meet the requirement: $1432,2143,2413,2431,3142,3214,3241,3412,3421,4132,4213,4231,4312,4321$.
When the number 5 is added to the above permutations, such as 51432, 15432, 14532, 14352, 14325 (the 3 crossed-out permutations contain the ascending sequence 145, which does not meet the requirement).
Considering all 14 permutations when adding the number 5, the number of permutations that meet the requirement is 42.