The array environment provides a very similar layout and typesetting to the amsmath package cases environment. In fact, amsmath defines cases exactly as an array with some stretch (an \arraystretch of 1.2) and a specific column alignment (@{}ll@{}). Consider the following minimal example displaying the two different styles:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\noindent Original \texttt{cases} example:
\[
X=%
\begin{cases}
0& \text{if $r-j$ is odd},\\
r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
\end{cases}
\]
\noindent Using the \texttt{array} environment:
\[
\renewcommand{\arraystretch}{1.2}%
X=\left\{%
\begin{array}{@{}ll@{}}
0& \text{if $r-j$ is odd},\\
r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
\end{array}
\right.
\]
\end{document}

Therefore, using the array environment, one could attempt:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\noindent Requested environment using \texttt{array}:
\[
\renewcommand{\arraystretch}{1.2}%
X=\left\{%
\begin{array}{@{}lll@{}}
0& & \text{if $r-j$ is odd},\\
k& +7& \text{if $r-j$ is even}. \\
& +1+2i& \\
& \binom{3}{n}
\end{array}
\right.
\]
\end{document}

The gap between the first two left-aligned columns can also be adjusted (using @{}l@{}ll@{}, say) if needed. The advantage of this approach is flexibility in terms of the array environment, while still retaining the visual equivalence with the cases environment; that is, a user-defined cases environment.
%in the\left\{%? – SoftTimur Aug 09 '11 at 02:37