Representation theory problem to solve.

Hi all,
I'm having to solve a few exercises from the book "Introduction to representation theory" (Etingof, Goldberg,...), and I am stuck on an exercise. In the book it's number 5.16.2:
The content $\displaystyle c(\lambda)$ of a Young diagram $\displaystyle \lambda$ is the sum $\displaystyle \sum_{j=1}^k\sum_{i=1}^{\lambda_{j}}(i-j)$, where $\displaystyle \lambda=(\lambda_{1},...,\lambda_{k})$ is a partition of $\displaystyle \lambda$. Let $\displaystyle C=\sum_{i<j}(ij)\in\mathbb{C}[S_{n}]$ be the sum of all transpositions. Show that $\displaystyle C$ acts on the Specht module $\displaystyle V_{\lambda}$ by multiplication by $\displaystyle c(\lambda)$.

I've been able to work this out with a few examples, but I don't really know how to get a proof.

Any help is much appreciated,
Thank you.