\\\\(\ extbf{Hint: Using probability.}\\\\)\\\\(\ extbf{Step 1: Find the probability that noted number is either 7 or 8.}\\\\) \\\\(\ ext{Probability of getting head}=\\dfrac{1}{2}\\\\) \\\\(\ ext{Probability of getting tail}=\\dfrac{1}{2}\\\\) \\\\(\ extbf{Probability of getting sum of 7 or 8 when dice is rolled.}\\\\) \\\\(P(E_1)=\\dfrac{6}{36}+\\dfrac{5}{36}=\\dfrac{11}{36}\\\\)\\\\(\ extbf{Step 2: Probability of getting sum of 7 or 8 in the shuffled card is} \\\\) \\\\(P(E_2)=\\dfrac{1}{9}+\\dfrac{1}{9}=\\dfrac{2}{9}\\\\)\\\\(\ extbf{Step 3: Now for the question.}\\Rightarrow\\ P(H\\cap E_1)+P(T\\cap E_2)\\\\) \\\\(Therefore, (H_1, E_1)\\ and\\ (T_1, E_2)\\ are\\ sets\\ of\\ independent\\ event.\\\\) \\\\(\\Rightarrow [P(H)\ imes P(E_1)]+[P(T)\ imes P(E_2)]\\\\)\\\\(\ extbf{Step 4: Substitute the value.}\\\\) \\\\(\\Rightarrow \\dfrac{1}{2}\ imes \\dfrac{11}{36}+\\dfrac{1}{2}\ imes \\dfrac{2}{9}\\\\) \\\\(\\Rightarrow \\dfrac{11}{72}+\\dfrac{8}{18}\\\\) \\\\(\\Rightarrow \\dfrac{19}{72}\\\\) \\\\(\ ext{Therefore,} \\ P(H\\cap E_1)+P(T\\cap E_2)=\\dfrac{19}{72}\\\\)\\\\(\ extbf{Hence the correct option is C. }\\\\)