Number of Elements in Partition

Theorem
Let $$S$$ be a set.

Let there be a partition on $$S$$ of $$n$$ subsets, each of which has $$m$$ elements.

Then $$\left|{S}\right| = n m$$.

Proof
Let the partition of $$S$$ be $$S_1, S_2, \ldots, S_n$$.

Then $$\forall k \in \left[{1 \,. \, . \, n}\right]: \left|{S_k}\right| = m$$.

By Power of an Element:

$$\sum_{k=1}^n \left|{S_k}\right| = n m$$

... and the result follows from the Fundamental Principle of Counting.