The number of values of $\mathrm{r}\in \{\mathrm{p},\mathrm{q},~\mathrm{p},~\mathrm{q}\}$ for which $\left(\right(\mathrm{p}\wedge \mathrm{q})\Rightarrow (\mathrm{r}\vee \mathrm{q}\left)\right)\wedge \left(\right(\mathrm{p}\wedge \mathrm{r})\Rightarrow \mathrm{q})$is a tautology is: 

$$\begin{gathered} \left(\right(\mathrm{p}\wedge \mathrm{q})\Rightarrow (\mathrm{r}\vee \mathrm{q}\left)\right)\wedge \left(\right(\mathrm{p}\wedge \mathrm{r})\Rightarrow \mathrm{q}) \\ \mathrm{p}\Rightarrow \mathrm{q}\text{ is equivalent to }~\mathrm{p}\vee \mathrm{q} \end{gathered}$$

$\begin{array}{r}(~(\mathrm{p}\wedge \mathrm{q})\vee (\mathrm{r}\vee \mathrm{q}\left)\right)\wedge (~(\mathrm{p}\wedge \mathrm{r}\left)\right)\vee \mathrm{q}\left)\right)\\ \Rightarrow (~\mathrm{p}\vee ~\mathrm{q}\vee \mathrm{r}\vee \mathrm{q})\wedge (~\mathrm{p}\vee ~\mathrm{r}\vee \mathrm{q})\\ \Rightarrow (~\mathrm{p}\vee \mathrm{r}\vee \mathrm{t})\wedge (~\mathrm{p}\vee ~\mathrm{r}\vee \mathrm{q})\\ \Rightarrow \left(\mathrm{t}\right)\wedge (~\mathrm{p}\vee ~\mathrm{r}\vee \mathrm{q})\end{array}$

$$\begin{gathered} \text{ For this to be tautology, }(~\mathrm{p}\vee ~\mathrm{r}\vee \mathrm{q})\text{ must be } \\ \text{ always true which follows for }r=\sim p\text{ or }r=q \end{gathered}$$

draw truth tables (or) given statement is a tautology for r can be $~\mathrm{p}\left(\text{ or }\right)\mathrm{q}$