​If two events $$A$$​ and $$B$$​ such that $$P(A∪B)=\frac{7}{8}$$​ and $$P(A∩B)=\frac{1}{4}$$​ and $$P(\bar{A})=\frac{5}{8}$$​, then $$P(\bar{A}∪\bar{B})=$$​ ? ​

Correct option is B

​$$\begin{aligned}&\text{Given } P(\overline{A}) = \frac{5}{8}, \text{ we know:} \\&P(A) = 1 - P(\overline{A}) = 1 - \frac{5}{8} = \frac{3}{8} \\\\&\textbf{Step 2: Find } P(B) \\&\text{Using the formula for the union of two events:} \\&P(A \cup B) = P(A) + P(B) - P(A \cap B) \\&\text{Substitute the known values:} \\&\frac{7}{8} = \frac{3}{8} + P(B) - \frac{1}{4} \\&\text{Convert } \frac{1}{4} \text{ to eighths:} \\&\frac{7}{8} = \frac{3}{8} + P(B) - \frac{2}{8} \\&\text{Simplify:} \\&\frac{7}{8} = \frac{1}{8} + P(B) \\&\text{Solve for } P(B): \\&P(B) = \frac{7}{8} - \frac{1}{8} = \frac{6}{8} = \frac{3}{4} \\\\&\textbf{Step 3: Find } P(\overline{B}) \\&P(\overline{B}) = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4} \\\\&\textbf{Step 4: Compute } P(\overline{A} \cup \overline{B}) \\&\text{Using De Morgan's Law:} \\&\overline{A \cup B} = \overline{A} \cap \overline{B} \\&\text{Thus:} \\&P(\overline{A} \cup \overline{B}) = P(\overline{A} \cap \overline{B}) = 1 - P(A \cap B) = 1 - \frac{1}{4} = \frac{3}{4}\end{aligned}$$​​