Number of Elements in Partition

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Theorem

Let $S$ be a set.

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


Then:

$\card S = n m$


Proof

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

Then:

$\forall k \in \set {1, 2, \ldots, n}: \card {S_k} = m$

By definition of multiplication:

$\displaystyle \sum_{k \mathop = 1}^n \card {S_k} = n m$

and the result follows from the Fundamental Principle of Counting.

$\blacksquare$


Sources