The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set $\{1, 2, 3\}$ is ________.

A $4$-digit number $d_1 d_2 d_3 d_4$ has digits in non-decreasing order from left to right if:

Since all digits belong to $\{1, 2, 3\}$, none of the digits can be $0$, so every such combination forms a valid $4$-digit number.

Selecting $4$ digits from a set of $3$ possible digits with repetition allowed uniquely determines a sorted non-decreasing sequence.

The number of ways to choose $k = 4$ items from $n = 3$ categories with repetition is given by the multiset coefficient:

Thus, the total number of such $4$-digit numbers is 15 [1].