= Solution
\b[Configuration count.] Choose the end-to-end direction as positive and let $N_R,N_L$ count rightward and leftward links. Then
$$
N_R+N_L=N,\qquad a(N_R-N_L)=L,
\qquad N_R=\frac{N+L/a}{2},\quad N_L=\frac{N-L/a}{2}.
$$
A configuration is determined by choosing the positions of its leftward links, so
$$
\boxed{\Omega(L)=\frac{N!}{N_R!N_L!}.}
$$
The counts must be nonnegative integers; in particular, allowed signed extensions occur in steps of $2a$. If distance means $|L|$ with neither direction selected, there are two corresponding sets for $L>0$, adding a factor of two. That constant does not affect the <tension>. There is no excluded-volume penalty in this one-dimensional idealized count.
\b[<Entropy> from the multiplicity.] Put $x=L/(Na)$, so $|x|<1$. Applying the leading <Stirling's approximation> to all three large factorials gives
$$
S=k_B\ln\Omega\simeq k_B[N\ln N-N_R\ln N_R-N_L\ln N_L]
$$
$$
\boxed{S(x)\simeq Nk_B\ln2-\frac{Nk_B}{2}
\left[(1+x)\ln(1+x)+(1-x)\ln(1-x)\right].}
$$
Terms subleading in $N$, including logarithmic factorial prefactors, have been omitted consistently. Expanding near zero extension gives
$$
S(L)\simeq Nk_B\ln2-\frac{k_BL^2}{2Na^2}
+O\left(Nk_Bx^4\right).
$$
Stretching reduces the number of available configurations and hence the <entropy>.
\b[<Tension> and the range of <Hooke's law>.] The model assigns no extension-dependent <internal energy>. At fixed <temperature>, the externally applied <tension> is $\mathcal F=(\partial F/\partial L)_T=-T(\partial S/\partial L)_T$. Differentiating the full leading <entropy> gives
$$
\boxed{\mathcal F=\frac{k_BT}{2a}\ln\frac{1+x}{1-x}
=\frac{k_BT}{a}\operatorname{artanh}x.}
$$
It is the external <force> needed to oppose the chain's restoring <entropic force>. Only when $|L|\ll Na$ can one put $\operatorname{artanh}x\simeq x$, yielding
$$
\boxed{\mathcal F\simeq\frac{k_BT}{Na^2}L,\qquad |L|\ll Na.}
$$
The paper's displayed linear law therefore requires this small-extension approximation, which is not explicitly stated in its premise $L<L_0$. It is not the exact <force> law up to full extension. The entropic spring constant grows with <temperature> because stronger thermal disorder makes a given reduction in configurational <entropy> cost more free <energy>. This is the <one-dimensional entropic chain force-extension relation>.
\b[A sparse number of backward links.] Write $q=N_L\ll N$, so $N_R=N-q$ and $L=a(N-2q)$. Exactly,
$$
\Omega=\frac{N(N-1)\cdots(N-q+1)}{q!}
=\frac{N^q}{q!}\prod_{j=0}^{q-1}\left(1-\frac jN\right).
$$
For fixed $q$ and large $N$, each numerator factor approaches $N$, giving
$$
\boxed{\Omega\simeq\frac{N^{N_L}}{N_L!}.}
$$
More precisely, the <logarithm> of the product is $-q(q-1)/(2N)+O(q^3/N^2)$. A small relative error in the multiplicity requires $q^2/N\ll1$; merely $q/N\ll1$ need not make the omitted product close to one. The weaker sparse-link condition does suffice for the leading <entropy> density and leading high-extension <tension>. These are <sparse backward links near full chain extension>.
\b[<Tension> in that limit.] In the overlap regime $1\ll q\ll N$, use Stirling also for $q!$:
$$
S\simeq k_B[q\ln N-q\ln q+q],\qquad
\frac{dq}{dL}=-\frac1{2a}.
$$
Thus
$$
\boxed{\mathcal F\simeq\frac{k_BT}{2a}\ln\frac N{N_L}.}
$$
The full leading <entropy> would give $(k_BT/(2a))\ln[(N-N_L)/N_L]$, which has the same dominant <logarithm> when $N_L/N\to0$.
\b[Approaching full extension.] The continuum <force> law diverges logarithmically as $N_L\to0$, or $L\to Na$. One must not literally apply Stirling to a zero factorial: at the discrete endpoint there is exactly one directed configuration, $\Omega=1$, and its <entropy> is zero. A continuous fixed-extension <derivative> is not defined at that endpoint.
An independent force-ensemble calculation makes the limiting physical statement precise. For applied <tension> $\mathcal F$, a link has <partition function> $z=2\cosh(\beta\mathcal Fa)$, giving the exact <mean> extension
$$
\boxed{\langle L\rangle=Na\tanh(\beta\mathcal Fa).}
$$
It approaches $Na$ only as $\mathcal F\to\infty$ at positive <temperature>, and inversion reproduces the logarithmic <force> law for <mean> extension. In a real chain, bond extension or failure eventually replaces the ideal inextensible-link model.
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