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]