jmc

Since there are so few numbers, we could simply list each of the $4\times 3\times 2\times 1=24$ combinations, but let's look at a more mathematical approach which we could also apply to larger sets of numbers.

We first consider how many of the numbers start with the digit $1.$ We have three more digits $(2,$ $3,$ and $4)$ to use. We can pick any of the three choices for the digit after the $1,$ and then either of the $2$ remaining choices for the third number, and finally, the $1$ remaining choice for the final number. Thus there are $3\times 2\times 1=6$ possibilities for numbers that begin with the digit $1.$ $($For completeness, these are: $1234,$ $1243,$ $1324,$ $1342,$ $1423,$ $1432.)$

The same reasoning can be used for numbers that begin with the digit $2.$ Therefore there are $6$ numbers which begin with $2.$ $($For completeness, these are: $2134,$ $2143,$ $2314,$ $2341,$ $2413,$ $2431.)$ After this, we have found a total of $12$ of the numbers in the list of $4$-digit integers with the digits $1,$ $2,$ $3,$ and $4.$

We also have $6$ different numbers which can be formed with a leading $3.$ This makes a total of $18$ different numbers, since we want the $15^{\text{th}}$ number, we can simply list these out in order from least to greatest, as specified in the problem.

$\bullet$ The $13^{\text{th}}$ number is $3124.$

$\bullet$ The $14^{\text{th}}$ number is $3142.$

$\bullet$ The $15^{\text{th}}$ number is $3214.$

$\bullet$ The $16^{\text{th}}$ number is $3241.$

$\bullet$ The $17^{\text{th}}$ number is $3412.$

$\bullet$ The $18^{\text{th}}$ number is $3421.$

Thus our answer is the $15\text{th}$ number, or $\boxed{3214}.$

Note that we could have stopped listing the numbers above once we got to the $15\text{th}$ number.