A system composed of $N$ quasi-independent particles follows Boltzmann statistics, and each particle can occupy any one of the three non-degenerate energy levels $-E, 0, +E$. The system is in contact with a heat reservoir at temperature $T$.
(1) What is the entropy of the system when $T = 0 \text{ K}$?
(2) What is the maximum entropy of the system?
(3) What is the minimum energy of the system?
(4) What is the partition function of the system?
(5) What is the most probable energy of the system?
(6) $c(T)$ is the specific heat capacity of the system:
$$
\int_{0}^{\infty} \frac{c(T)}{T} \mathrm{d} T = ?
$$

Options:
A. $$
N k \mathrm{l n} 10
$$
$$
N E / k
$$
$$
- N E^3
$$
$$
(1-\mathrm{e}^{E / k T}+\mathrm{e}^{-E / k T} )^{N}
$$
$$
-\; {\frac{5 N E \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)} {3+3 \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 3
$$
B. $$
N k \mathrm{l n} 7
$$
$$
N k T
$$
$$
- N E / 3
$$
$$
( \mathrm{e}^{E / 2k T}+1+\mathrm{e}^{-E / 2k T} )^{N}
$$
$$
-\; {\frac{N E \operatorname{t a n h} \Bigl( {\frac{E} {k T}} \Bigr)} {2+2 \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 8
$$
C. $$
N k \mathrm{l n} 2
$$
$$
0
$$
$$
- N E / 2
$$
$$
( \mathrm{e}^{E / k T}-1+\mathrm{e}^{-E / k T} )^{N}
$$
$$
-\; {\frac{N E \operatorname{t a n h} \Bigl( {\frac{E} {k T}} \Bigr)} {1+ \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 4
$$
D. $$
N k \mathrm{l n} 11
$$
$$
0
$$
$$
- N E^{3/2}
$$
$$
( \mathrm{e}^{E / kT}+2+\mathrm{e}^{-E / kT} )^{N}
$$
$$
-\; {\frac{6 N E \operatorname{t a n h} \Bigl( {\frac{E} {k T}} \Bigr)} {4+3 \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 4
$$
E. $$
N k \mathrm{l n} 1
$$
$$
0T
$$
$$
-E
$$
$$
(1+ \mathrm{e}^{-E / 3k T}+\mathrm{e}^{E / 3k T})^{N}
$$
$$
-\; {\frac{8 N E \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)} {6+ \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 3
$$
F. $$
N k \mathrm{l n} 12
$$
$$
N 0
$$
$$
- 2N E
$$
$$
( \mathrm{e}^{E^2 / k T}+1+\mathrm{e}^{-E^2 / k T} )^{N}
$$
$$
-\; {\frac{7 N E \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)} {5+ \operatorname{t a n h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 5
$$
G. $$
N k \mathrm{l n} 1
$$
$$
N / k T
$$
$$
- 2 N E
$$
$$
(1+\mathrm{e}^{-2E / k T}+\mathrm{e}^{2E / k T})^{N}
$$
$$
\; {\frac{N E \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)} {1-2 \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 5
$$
H. $$
N k \mathrm{l n} 9
$$
$$
0
$$
$$
- N E / 4
$$
$$
(\mathrm{e}^{2E / k T}+1+\mathrm{e}^{-2E / k T})^{N}
$$
$$
-\; {\frac{4 N E \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)} {3+2 \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 2
$$
I. $$
N k \mathrm{l n} 5
$$
$$
0
$$
$$
- N E^2
$$
$$
( \mathrm{e}^{-E / k T}+2+\mathrm{e}^{E / k T} )^{N}
$$
$$
-\; {\frac{3 N E \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)} {2+ \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 6
$$
J. $$
0
$$
$$
N k \mathrm{l n} 3
$$
$$
- N E
$$
$$
( \mathrm{e}^{E / k T}+1+\mathrm{e}^{-E / k T} )^{N}
$$
$$
-\; {\frac{2 N E \operatorname{s i n h} \Bigl( {\frac{E} {k T}} \Bigr)} {1+2 \operatorname{c o s h} \Bigl( {\frac{E} {k T}} \Bigr)}}
$$
$$
N k \mathrm{l n} 3
$$
Answer: J