  Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, a subset
$\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of
$\mathcal{F}$ agree on some element of $\Omega$. The \emph{intersection
density} of $G$, denoted by $\rho(G)$, is the maximum of the rational number
$|\mathcal{F}|\left(\frac{|G|}{|\Omega|}\right)^{-1}$ when $\mathcal{F}$ runs
through all intersecting sets in $G$. In this paper, we prove that if $G$ is
the group $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the
$k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $\rho(G)=1$. Our
proof relies on the representation theory of the symmetric group and the ratio
bound.
