To expand \((a + 1)^3\) using Pascal's Triangle, we first identify the coefficients from the third row of Pascal's Triangle. The rows of Pascal's Triangle start with row 0 at the top:
- Row 0: \(1\)
- Row 1: \(1 \quad 1\)
- Row 2: \(1 \quad 2 \quad 1\)
- Row 3: \(1 \quad 3 \quad 3 \quad 1\)
The coefficients for the expansion of \((a + b)^n\) are found in row \(n\). Since we are expanding \((a + 1)^3\), we use row 3: \(1\), \(3\), \(3\), \(1\).
The expansion is given by:
\[ (a + 1)^3 = 1 \cdot a^3 + 3 \cdot a^2 \cdot 1 + 3 \cdot a \cdot 1^2 + 1 \cdot 1^3 \]
This simplifies to:
\[ = a^3 + 3a^2 + 3a + 1 \]
So, the expansion of \((a + 1)^3\) is:
\[ a^3 + 3a^2 + 3a + 1 \]