= Solution
For shape $(n-2,2)$ with $n\ge6$, there are exactly two removable <rim hooks> of size two: the bottom row, leaving $(n-2)$, and the final two cells of the first row, leaving $(n-4,2)$. Both have height zero. A vertical pair cannot be removed while leaving a <Young diagram> because the first row is longer than the second by at least two.
By the <Murnaghan–Nakayama rule> and part (b),
$$
\begin{aligned}
\chi^{(n-2,2)}(2,1^{n-2})
&=1+f^{(n-4,2)}\\
&=1+\binom{n-2}{2}-\binom{n-2}{1}
=\boxed{\frac{(n-4)(n-3)}2}.
\end{aligned}
$$
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