Let $\mathrm{P}\left(\mathrm{x}\right)={\mathrm{x}}^3-8{\mathrm{x}}^2+\mathrm{cx}-\mathrm{d}$ and be a polynomial with real coefficients and with all its roots being distinct positive integers. Then number of possible value of c is ____.

We have, ${\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3=8$
            $\begin{array}{l}{\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3=\mathrm{d}\\ {\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{x}}_3{\mathrm{x}}_1=\mathrm{c}\end{array}$
Possible roots 1,2,5 or 1,3,4
$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{d}=10\text{ or }\mathrm{d}=12\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}=2+10+5=17\text{ or }3+12+4=19\end{array}$
Hence, $\mathrm{d}=10\text{ and }\mathrm{c}=17\text{ or }\mathrm{d}=12\text{ and }\mathrm{c}=19$