(a)     Simplify the difference of binomial coefficients

$$\left( {\begin{array}{*{20}{c}}   n \\   3 \end{array}} \right) - \left( {\begin{array}{*{20}{c}}   {2n} \\   2 \end{array}} \right),{\text{ where }}n \geqslant 3.$$

(b)     Hence, solve the inequality

$$\left( {\begin{array}{*{20}{c}}   n \\   3 \end{array}} \right) - \left( {\begin{array}{*{20}{c}}   {2n} \\   2 \end{array}} \right) > 32n,{\text{ where }}n \geqslant 3.$$

(a)     the expression is

$\frac{{n!}}{{(n - 3)!3!}} - \frac{{(2n)!}}{{(2n - 2)!2!}}$     (A1)

$\frac{{n(n - 1)(n - 2)}}{6} - \frac{{2n(2n - 1)}}{2}$     M1A1

$= \frac{{n({n^2} - 15n + 8)}}{6}{\text{ }}\left( { = \frac{{{n^3} - 15{n^2} + 8n}}{6}} \right)$     A1

(b)     the inequality is

$\frac{{{n^3} - 15{n^2} + 8n}}{6} > 32n$

attempt to solve cubic inequality or equation     (M1)

${n^3} - 15{n^2} - 184n > 0\,\,\,\,\,n(n - 23)(n + 8) > 0$

$n > 23\,\,\,\,\,(n \geqslant 24)$     A1

[6 marks]

Part(a) - Although most understood the notation, few knew how to simplify the binomial coefficients.

Part(b) - Many were able to solve the cubic, but some failed to report their answer as an integer inequality.