The particles in a Fermi gas have three non-degenerate energy levels 0, $\varepsilon$, and $2\varepsilon$. If the gas can have at most $2$ particles, what are the grand partition function and the average particle number of the gas ?

Options:
A. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/k T )]}}}\\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+2 \lambda ]
$$
B. $$
\begin{array} {l} {{{1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {\Xi} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-\varepsilon/ k T ) ]
$$
C. $$
\begin{array} {l} {{{2\varepsilon1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-\varepsilon/ k T ) ]
$$
D. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1+\operatorname{e x p} (-2\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2\varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-\varepsilon/ k T ) ]
$$
E. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-2\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1 + 2\lambda \operatorname{e x p} (-\varepsilon/kT)]
$$
F. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )-\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+2\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-2\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-\varepsilon/ k T ) ]
$$
G. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\frac{\varepsilon}{k T }) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+ \lambda\operatorname{e x p} (-\varepsilon/ k T ) ]
$$
H. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+2\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-2\varepsilon/ k T ) ]
$$
I. $$
\begin{array} {l} {{{\varepsilon1+\lambda[ 1-\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-2\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-\varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-\varepsilon/ k T ) ]
$$
J. $$
\begin{array} {l} {{{\varepsilon2+\lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ {{{\quad+\lambda^{2} \mathrm{e x p} (-\varepsilon/ k T ) [ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ]}}} \\ \end{array}
$$
$$\frac{1} {E} \lambda[ 1+\operatorname{e x p} (-\varepsilon/ k T )+\operatorname{e x p} (-2 \varepsilon/ k T ) ] \cdot[ 1+2 \lambda\operatorname{e x p} (-2\varepsilon/ k T ) ]
$$
Answer: B