= Solution
Let $H$ be the $r$-<uniform hypergraph> and use <strong saturation of a uniform hypergraph> in its immediate-completion sense: each missing edge $e$ lies in an $(r+t)$-set $T$ whose other $r$-subsets are already edges of $H$. The argument allows existing <hypergraph cliques>. Take $n\geq r+t$, the usual parameter range; if $r\leq n<r+t$, strong saturation forces every possible edge to be present, and the inequality is immediate whenever its binomial terms are defined. The case $t=0$ has the trivial lower bound zero.
For $t\geq1$, choose distinct real numbers $z_1,\ldots,z_n$ and form the $t\times n$ <Vandermonde matrix> $M$ whose $i$th column is $(1,z_i,\ldots,z_i^{t-1})^T$. Every $t$ columns are <linearly independent> by the <Vandermonde determinant>. Set $U=\ker M\subseteq\mathbb R^n$, so $\dim U=n-t$, and work in the <quotient vector space>
$$
W=\bigwedge^r\mathbb R^n\big/\bigwedge^rU,\qquad\dim W=\binom nr-\binom{n-t}r.
$$
Here the inclusion of $\bigwedge^rU$ is injective: extend a basis of $U$ to a basis of $\mathbb R^n$ and use the resulting wedge bases of the <exterior powers>. Write $w_e$ for the image in $W$ of the coordinate wedge indexed by an $r$-set $e$. All these images span $W$.
For any $(r+t)$-set $T$, let $V_T$ be its coordinate subspace. The restriction of $M$ has rank $t$, so $U\cap V_T$ has dimension $r$. Wedge a basis of this intersection to obtain a nonzero element
$$
\omega_T=\sum_{\substack{e\subseteq T\\|e|=r}}c_e\,e_e\in\bigwedge^rU.
$$
Every coefficient $c_e$ is nonzero. Indeed, projection of $U\cap V_T$ onto the $e$ coordinates is an isomorphism: a vector in its kernel is supported on $T\setminus e$, whose $t$ columns of $M$ are independent. The coefficient is the determinant of that projection in the chosen bases. Passing to $W$ therefore gives a <linear dependence>
$$
\sum_{\substack{e\subseteq T\\|e|=r}}c_e w_e=0\qquad(c_e\ne0).
$$
If $e$ is missing from $H$, take its strong-saturation witness $T$. The dependence expresses $w_e$ in the span of the $w_f$ for the other edges $f\subseteq T$, all of which are present. Thus the vectors indexed by the present edges span all of $W$. The <exterior-power quotient bound for strong hypergraph saturation> follows:
$$
\boxed{|E(H)|\geq\dim W=\binom nr-\binom{n-t}r.}
$$
The bound is sharp: include precisely the $r$-sets meeting one fixed $t$-set $S$. A missing edge $e$ is disjoint from $S$, and every other $r$-subset of $e\cup S$ meets $S$, so adding $e$ immediately completes a <hypergraph clique> of order $r+t$. The number of included edges is exactly the displayed bound.
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