(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.