(c) Using Pascal's triangle, expand the following binomials: (i) \( (2 a+1)^{3} \)

To expand the binomial \( (2a + 1)^3 \) using Pascal's triangle, we first need to identify the coefficients from the triangle for \( n = 3 \). The coefficients for \( n = 3 \) in Pascal's triangle are: \[ 1, 3, 3, 1 \] Using these coefficients, we can expand \( (2a + 1)^3 \) as follows: \[ (2a + 1)^3 = \sum_{k=0}^{3} \binom{3}{k} (2a)^{3-k} (1)^k \] This can be written out as: \[ = \binom{3}{0} (2a)^3 (1)^0 + \binom{3}{1} (2a)^2 (1)^1 + \binom{3}{2} (2a)^1 (1)^2 + \binom{3}{3} (2a)^0 (1)^3 \] Now substituting the coefficients: \[ = 1 \cdot (2a)^3 + 3 \cdot (2a)^2 \cdot 1 + 3 \cdot (2a) \cdot 1^2 + 1 \cdot 1^3 \] Calculating each term: 1. \( (2a)^3 = 8a^3 \) 2. \( 3 \cdot (2a)^2 = 3 \cdot 4a^2 = 12a^2 \) 3. \( 3 \cdot (2a) = 6a \) 4. \( 1 \) Putting it all together: \[ (2a + 1)^3 = 8a^3 + 12a^2 + 6a + 1 \] Thus, the expansion of \( (2a + 1)^3 \) is: \[ \boxed{8a^3 + 12a^2 + 6a + 1} \]