1. **State the problem:** Simplify the Boolean expression: $$\overline{A}BC\overline{D} + \overline{A}BC\overline{D} + \overline{A}BC\overline{O} + \overline{A}BC\overline{D} + \overline{A}B\overline{E}D + \overline{A}BC\overline{O}$$ 2. **Identify variables and notation:** Here, $\overline{A}$ means NOT A, $B$, $C$, $D$, $E$ are variables, and $\overline{D}$ means NOT D, etc. The symbol $\overline{O}$ likely is a misread or typo; assuming it means $\overline{D}$ or $\overline{O}$ is a variable negated. For clarity, treat $\overline{O}$ as $\overline{D}$ or a distinct variable negated. 3. **Combine like terms:** Notice repeated terms: - $\overline{A}BC\overline{D}$ appears multiple times. - $\overline{A}BC\overline{O}$ appears twice. Using idempotent law: $X + X = X$, so duplicates can be removed. 4. **Simplify expression:** $$\overline{A}BC\overline{D} + \overline{A}BC\overline{O} + \overline{A}B\overline{E}D$$ 5. **Factor common terms:** - Factor $\overline{A}B$: $$\overline{A}B(C\overline{D} + C\overline{O} + \overline{E}D)$$ 6. **Simplify inside parentheses:** - $C\overline{D} + C\overline{O} = C(\overline{D} + \overline{O})$ So expression is: $$\overline{A}B\bigl(C(\overline{D} + \overline{O}) + \overline{E}D\bigr)$$ 7. **Final simplified form:** $$\boxed{\overline{A}B\bigl(C(\overline{D} + \overline{O}) + \overline{E}D\bigr)}$$ This is the simplest form given the variables and terms. **Note:** If $\overline{O}$ is a typo or meant to be $\overline{D}$, then further simplification is possible by combining terms.