[Number Theory]
Problem: Compute the number of five-digit integers $\underline{M} \underline{A} \underline{R} \underline{T} \underline{Y}$, with all digits distinct, such that $M>A>R$ and $R<T<Y$.

Solution: There are $\left(\begin{array}{c}10 \\ 5\end{array}\right)=252$ ways to choose the values of the digits $M, A, R, T, Y$, without restrictions. Because $R$ is fixed as the least of the digits and because $T<Y$, it suffices to find the number of ways to choose $M$ and $A$. Once $M$ and $A$ are chosen, the other three digits are uniquely determined. There are $\left(\begin{array}{l}4 \\ 2\end{array}\right)=6$ ways to select $M, A$. Thus the number of five-digit integers of the type described is $252 \cdot 6=\mathbf{1 5 1 2}$.

Final answer: 1512