Find the number of positive integers that satisfy both the following conditions:
* Each digit is a $1$ or a $3$ or a $5$
* The sum of the digits is $10$
Let the number be $abcde$ where $a,b,c,d,e$ represent the digits (with $a$ representing the leftmost digit). We have the two equations \begin{align*}
a+b+c+d+e&=10 \\ \quad a,b,c,d,e &\in \{1,3,5\}.
\end{align*} We can find the number of solutions to this equation using \emph{sticks and stones}. Picture the $10$ objects arranged in a line. Now place four dividing sticks amongst the objects. These dividing sticks will divide the $10$ objects (stones) into $5$ parts. For example, ||*||*|*|**| represents the number $a=2, b = 0, c = 1, d = 2, e = 5.$ The five parts represented by $a, b, c, d, e$ add up to $10.$ There are $\binom{10+5-1}{5-1}=\binom{14}{4}=1001$ ways to arrange these sticks and stones, thus $\boxed{1001}$ integers satisfy the conditions given in the problem.