For a certain molecule, the three lowest energy levels are $E_{1} = 0, E_{2} = \varepsilon, E_{3} = 10\varepsilon$. It is proven that at sufficiently low temperatures, only the energy levels $E_{1}$ and $E_{2}$ are occupied. What is the average energy $E$ of the molecule at temperature $T$ ? What are the contributions of these energy levels to the molar heat capacity $c_{v}$ ?

Options:
A. $$
\frac{\varepsilon( \mathrm{e}^{-3\varepsilon/ k T}+4 2 \mathrm{e}^{-1 0 \varepsilon/ k T} )} {1+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-4 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{2} ( \mathrm{e}^{-\beta t}+4 1 \mathrm{e}^{-1 0 \beta t}+6 2 \mathrm{e}^{-1 1 \beta t} )} {( 5+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 0 \beta t} )^{2}} \beta^{2}
$$
B. $$
\frac{3\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+3 \mathrm{e}^{-1 0 \varepsilon/ k T} )} {1+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-5 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{2} ( \mathrm{e}^{-3\beta t}+1 2 \mathrm{e}^{-9 \beta t}+8 0 \mathrm{e}^{-2 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-2 0 \beta t} )^{2}} \beta^{3}
$$
C. $$
\frac{\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+1 0 \mathrm{e}^{-1 0 \varepsilon/ k T} )} {1+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-1 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{2} ( \mathrm{e}^{-\beta t}+1 0 0 \mathrm{e}^{-1 0 \beta t}+8 1 \mathrm{e}^{-1 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 0 \beta t} )^{2}} \beta^{2}
$$
D. $$
\frac{6\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+3 \mathrm{e}^{-1 2 \varepsilon/ k T} )} {1+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-1 0 \varepsilon/ k T}}
$$
$$
\frac{2 R \varepsilon^{2} ( \mathrm{e}^{-\beta t}+2 8 \mathrm{e}^{-1 0 \beta t}+9 3 \mathrm{e}^{-1 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 0 \beta t} )^{4}} \beta^{2}
$$
E. $$
\frac{4\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+2 5 \mathrm{e}^{-1 1 \varepsilon/ k T} )} {2+\mathrm{e}^{-2\varepsilon/ k T}+\mathrm{e}^{-8 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{2} ( \mathrm{e}^{-\beta t}+5 0 \mathrm{e}^{-1 0 \beta t}+7 7 \mathrm{e}^{-2 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 2 \beta t} )^{2}} \beta^{4}
$$
F. $$
\frac{\varepsilon^{2}( \mathrm{e}^{-\varepsilon/ k T}+9 \mathrm{e}^{-1 0 \varepsilon/ k T} )} {2+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-1 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon( \mathrm{e}^{-\beta t}+9 9 \mathrm{e}^{-1 0 \beta t}+3 \mathrm{e}^{-1 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 0 \beta t} )^{3}} \beta^{2}
$$
G. $$
\frac{\varepsilon( \mathrm{e}^{-2\varepsilon/ k T}+5 \mathrm{e}^{-1 0 \varepsilon/ k T} )} {3+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-1 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon( \mathrm{e}^{-\beta t}+2 0 \mathrm{e}^{-1 0 \beta t}+9 0 \mathrm{e}^{-1 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-3 0 \beta t} )^{2}} \beta^{2}
$$
H. $$
\frac{2\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+3 7 \mathrm{e}^{-2 0 \varepsilon/ k T} )} {1+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-1 9 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{2} ( \mathrm{e}^{-\beta t}+7 7 \mathrm{e}^{-2 0 \beta t}+5 5 \mathrm{e}^{-1 1 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-9 0 \beta t} )^{2}} \beta^{6}
$$
I. $$
\frac{5\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+1 5 \mathrm{e}^{-1 0 \varepsilon/ k T} )} {2+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-3 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{2} ( \mathrm{e}^{-\beta t}+6 1 \mathrm{e}^{-1 0 \beta t}+2 2 \mathrm{e}^{-7 1 \beta t} )} {( 3+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 0 \beta t} )^{2}} \beta^{8}
$$
J. $$
\frac{7\varepsilon( \mathrm{e}^{-\varepsilon/ k T}+8 5 \mathrm{e}^{-1 9 \varepsilon/ k T} )} {1+\mathrm{e}^{-\varepsilon/ k T}+\mathrm{e}^{-1 0 \varepsilon/ k T}}
$$
$$
\frac{R \varepsilon^{1.5} ( \mathrm{e}^{-3\beta t}+1 3 \mathrm{e}^{-9 \beta t}+8 0 \mathrm{e}^{-3 0 \beta t} )} {( 1+\mathrm{e}^{-\beta t}+\mathrm{e}^{-1 0 \beta t} )^{2}} \beta^{2}
$$
Answer: C