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Here's another proof.

If $G$ acts transitively on $X$, then the permutation representation of $G$ is induced from the permutation representation of an arbitrary the stabilizer $S$. S$ of an arbitrary point. As a result, $\langle \chi , \chi \rangle$ counts the number of orbits of $S$ on $X$.

In your situation $G=S_n$ is acting transitively on $X=\{1,\ldots,n\}$. Let $S$ be the stabilizer of $n$; this is pretty much $S_{n-1} \subset S_n$, which has two orbits on $X$: $\{1,\ldots, n-1\}$ and $\{n\}$.
show/hide this revision's text 1

Here's another proof.

If $G$ acts transitively on $X$, then the permutation representation of $G$ is induced from the permutation representation of an arbitrary stabilizer $S$. As a result, $\langle \chi , \chi \rangle$ counts the number of orbits of $S$ on $X$.

In your situation $G=S_n$ is acting transitively on $X=\{1,\ldots,n\}$. Let $S$ be the stabilizer of $n$; this is pretty much $S_{n-1} \subset S_n$, which has two orbits on $X$: $\{1,\ldots, n-1\}$ and $\{n\}$.