Find the number of distinct integers with non-decreasing digits formed from one or more of the digits $2, 2, 3, 3, 4, 5, 5, 5, 6, 7$ Suppose integers are formed by taking one or more digits from the 
following: $2, 2, 3, 3, 4, 5, 5, 5, 6, 7$. For example, $355$ is a possible choice while $44$ is not. Find the number of distinct integers that can be formed in which the digits are non-decreasing.  

An integer with non-decreasing digits selected from the digits $2, 2, 3, 3, 4, 5, 5, 5, 6, 7$ is a selection of one of more digits of the string $2233455567$.  For instance, the selection $22\color{blue}{33}4\color{blue}{5}55\color{blue}{67}$ corresponds to the integer $33567$.  
Since the digits of the integer are non-decreasing, a particular integer is completely determined by the number of times each digit is selected.  We can select a particular digit up to the number of times it occurs in the string.  For instance, $2$ appears twice in the string $2233455567$, so we can select the digit $2$ either $0$, $1$, or $2$ times, giving us three choices for the number of occurrences of the digit $2$ in the integer.  By similar reasoning, we have three choices for the number of occurrences of the digit $3$ in the integer, four choices for the number of occurrences of the digit $5$ in the integer, and two choices for the number of occurrences of each of the digits $4$, $6$, and $7$.  Hence, it would appear that the number of integers we can form with the given digits is 
$$3 \cdot 3 \cdot 2 \cdot 4 \cdot 2 \cdot 2 = 288$$
However, the integer must be formed using at least one digit, and we have counted the empty string.  Therefore, the number of distinct integers with non-decreasing digits that can be formed from permutations of $2, 2, 3, 3, 4, 5, 5, 5, 6, 7$ is $288 - 1 = 287$. 
