= Solution
Compare the coefficient of $\{t\}$ in $C e(t)$. For a transposition $\tau$, a summand $\tau\pi\{t\}$ with $\pi\in C(t)$ equals $\{t\}$ exactly when $\tau\pi\in R(t)$, equivalently $\tau\in R(t)C(t)$. Part b(i) then leaves two cases: a row transposition contributes $+1$ through $\pi=1$, while a column transposition contributes $-1$ through $\pi=\tau$. Every other transposition contributes zero.
The coefficient is consequently the number of row transpositions minus the number of column transpositions, which part b(ii) identifies with $\sum_{(i,j)\in\lambda}c_{i,j}(\lambda)$. The coefficient of $\{t\}$ in $c(\lambda)e(t)$ is $c(\lambda)$, so
$$
c(\lambda)=\sum_{(i,j)\in\lambda}c_{i,j}(\lambda).
$$
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