Let universe be completely made from hydrogen. And also we have redshift $z= 6$. with Hubble constant $H_{0} = 2.1941747572815535\times 10^{-18}\:\mathrm{s}^{-1}$. We also know that density of the universe $\rho =\frac{3H^2}{8\pi G}$ ($G= 6.67\times 10^{-11}\:\mathrm{m^3\ kg^{-1}\ s^{-2}}$).
Is the following approach right to find the number density of hydrogen in universe?
First we calculate density $\rho =\frac{3H^2}{8\pi G}$, then we have $1+z = \sqrt{\frac{1+v/c}{1-v/c}}$. From that we get $v = H\times R$ and from this we calculate $R$, the radius of observable universe. With that, we calculate volume of universe $V = (4\pi/3)R^3$ and we get Mass $M=\rho V$. Then we calculate number of Hydrogen atoms with $N = M/m_{\mathrm{hydrogen}}$. Now we have number density of hydrogen $\rho = N/V$.
Is this solution right? Or we must solve it another way?
The source of question is National Olympiad of Iran which is not available in English so the above is my translation for the question.