(a) The binomial expansion of \((1 + x)^5\) is given by:

\(1 + 5x + \frac{5 \times 4}{2!}x^2 = 1 + 5x + 10x^2\).

(b) The binomial expansion of \((1 - 2x)^6\) is given by:

\(1 - 6(2x) + \frac{6 \times 5}{2!}(2x)^2 = 1 - 12x + 60x^2\).

(c) To find the coefficient of \(x^2\) in \((1 + x)^5 (1 - 2x)^6\), consider the products that result in \(x^2\):

- \(10x^2\) from \((1 + 5x + 10x^2) \times 1\)

- \(-60x^2\) from \(1 \times 60x^2\)

- \(60x^2\) from \(5x \times -12x\)

The coefficient of \(x^2\) is:

\(10 - 60 + 60 = 10\).