To expand \((a + b)^3\) using Pascal's Triangle, we start by identifying the row that corresponds to \(n = 3\).
The fourth row of Pascal's Triangle (since we start counting from row 0) is: \[ 1, 3, 3, 1 \]
These coefficients will be used in the expansion of \((a + b)^3\). The general form of the binomial expansion is given by: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For \(n = 3\), this becomes: \[ (a + b)^3 = \binom{3}{0} a^{3} b^{0} + \binom{3}{1} a^{2} b^{1} + \binom{3}{2} a^{1} b^{2} + \binom{3}{3} a^{0} b^{3} \]
Substituting the coefficients from Pascal's Triangle: \[ (a + b)^3 = 1 \cdot a^3 + 3 \cdot a^2 b + 3 \cdot a b^2 + 1 \cdot b^3 \]
Thus, the expansion of \((a + b)^3\) is: \[ a^3 + 3a^2b + 3ab^2 + b^3 \]
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