1. **Problem statement:** Simplify the Boolean expression $$M \cdot N \cdot P + I \cdot M \cdot P + I \cdot M \cdot N + I \cdot M \cdot \overline{N} + I \cdot N \cdot \overline{P}$$. 2. **Recall Boolean algebra rules:** - Idempotent law: $$A + A = A$$ - Distributive law: $$A \cdot (B + C) = A \cdot B + A \cdot C$$ - Complement law: $$A + \overline{A} = 1$$ - Absorption law: $$A + A \cdot B = A$$ 3. **Group terms to factor common variables:** $$M \cdot N \cdot P + I \cdot M \cdot P + I \cdot M \cdot N + I \cdot M \cdot \overline{N} + I \cdot N \cdot \overline{P}$$ Group terms with common factors: $$= M \cdot N \cdot P + I \cdot M \cdot P + I \cdot M \cdot N + I \cdot M \cdot \overline{N} + I \cdot N \cdot \overline{P}$$ 4. **Factor $$I \cdot M$$ from the middle three terms:** $$= M \cdot N \cdot P + I \cdot M \cdot (P + N + \overline{N}) + I \cdot N \cdot \overline{P}$$ Since $$P + N + \overline{N} = P + 1 = 1$$ (because $$N + \overline{N} = 1$$), this simplifies to: $$= M \cdot N \cdot P + I \cdot M + I \cdot N \cdot \overline{P}$$ 5. **Now, observe the first and last terms:** $$M \cdot N \cdot P + I \cdot N \cdot \overline{P} + I \cdot M$$ 6. **No further common factors between $$M \cdot N \cdot P$$ and $$I \cdot N \cdot \overline{P}$$, so keep as is. The simplified expression is:** $$\boxed{I \cdot M + M \cdot N \cdot P + I \cdot N \cdot \overline{P}}$$ This is the simplest form combining terms and factoring where possible.