HMMT November, 2018

Solution:

We use the factorial base, where we denote
$$\left(d_{n} \ldots d_{1}\right)_{*}=d_{n} \times n!+\cdots+d_{1} \times 1!$$
The representation of $2018_{10}$ is $244002_{*}$ and the representation of $720_{10}$ is $100000_{*}$. The largest nice number less than $244002_{*}$ is $243321_{*}$. Notice that for the digit $d_{i}$ of a nice number, we can vary its value from 1 to $i$, while for a generic number in the factorial base, $d_{i-1}$ can vary from 0 to $i-1$. Hence we can map nice numbers to all numbers by truncating the last digit and reducing each previous digit by 1, and likewise reverse the procedure by increasing all digits by 1 and adding 1 at the end. Furthermore, this procedure preserves the ordering of numbers. Applying this procedure to $243321_{*}$ gives $13221_{*}$. We count from $0_{*}$ to $13221_{*}$ (since the first nice number is $1_{*}$), to get an answer of
$$13221_{*}+1=210$$