Counting a Partition into Blocks of Equal Cardinality

Let $X$ be a set, let $r,t\in\mathbb{N}$ be natural numbers, and let $[r]$ be the initial segment determined by $r$. Suppose that for each $i\in[r]$ a subset $B_i\subseteq X$ is given such that:

1. every $x\in X$ lies in $B_i$ for some $i\in[r]$;
2. $B_i\cap B_{i'}=\emptyset$ whenever $i,i'\in[r]$ and $i\ne i'$;
3. each $B_i$ has $t$ elements.

Then $X$ has $t\cdot r$ elements.