Express the binomial coefficient \(\left( \begin{gathered}
3n + 1 \hfill \\
3n - 2 \hfill \\
\end{gathered} \right)\) as a polynomial in \(n\).

Hence find the least value of \(n\) for which \(\left( \begin{gathered}
3n + 1 \hfill \\
3n - 2 \hfill \\
\end{gathered} \right) > {10^6}\).

\(\left( \begin{gathered}
3n + 1 \hfill \\
3n - 2 \hfill \\
\end{gathered} \right) = \frac{{\left( {3n + 1} \right){\text{!}}}}{{\left( {3n - 2} \right){\text{!}}3{\text{!}}}}\)     (M1)

\( = \frac{{\left( {3n + 1} \right)3n\left( {3n - 1} \right)}}{{3{\text{!}}}}\)     A1

\( = \frac{9}{2}{n^3} - \frac{1}{2}n\) or equivalent     A1

[3 marks]

attempt to solve \( = \frac{9}{2}{n^3} - \frac{1}{2}n > {10^6}\)     (M1)

\(n > 60.57 \ldots \)     (A1)

Note: Allow equality.

\( \Rightarrow n = 61\)     A1

[3 marks]