Ideal gas Helmholtz energy function

From equations

$Q_{NVT}={\frac {1}{N!}}\left({\frac {V}{\Lambda ^{3}}}\right)^{N}$

for the canonical ensemble partition function for an ideal gas, and

$\left.A\right.=-k_{B}T\ln Q_{NVT}$

for the Helmholtz energy function, one has

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=-k_{B}T\left(\ln {\frac {1}{N!}}+N\ln {\frac {V}{\Lambda ^{3}}}\right)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-k_BT\left(-\ln N! + N\ln\frac{VN}{\Lambda^3N}\right)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-k_BT\left(-\ln N! + N\ln\frac{N}{\Lambda^3 \rho}\right)}

using Stirling's approximation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle =-k_{B}T\left(-N\ln N+N+N\ln N-N\ln \Lambda ^{3}\rho \right)}

one arrives at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} is the de Broglie thermal wavelength and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant.