Solution:

Answer: $39$

Note that $1 = 0!$, $2 = 0! + 1!$, $3 = 0! + 2!$, and $4 = 0! + 1! + 2!$. These are the only numbers less than $6$ that can be written as the sum of factorials. The only other factorials less than $240$ are $3! = 6$, $4! = 24$, and $5! = 120$. So a positive integer less than or equal to $240$ can only contain $3!$, $4!$, $5!$, and/or one of $1, 2, 3$, or $4$ in its sum. If it contains any factorial larger than $5!$, it will be larger than $240$. So a sum less than or equal to $240$ will either include $3!$ or not ($2$ ways), $4!$ or not ($2$ ways), $5!$ or not ($2$ ways), and add an additional $0, 1, 2, 3$ or $4$ ($5$ ways). This gives $2 \cdot 2 \cdot 2 \cdot 5 = 40$ integers less than $240$. However, we want only positive integers, so we must not count $0$. So there are $39$ such positive integers.