TITLE: Calculating number of non-zero elements in a lower triangular matrix
QUESTION [6 upvotes]: Given a lower triangular matrix $M$ of size $m$ by $n$, is there an equation for the number of elements in this matrix that can be non-zero?  What if the matrix is strictly lower triangular?

REPLY [2 votes]: I personally like Girardis Solution. The same solution can be derived (motivated by the answer of victor_v) by appling k-combination with repetitions. This allows to include the elements in the diagonal.
$$\binom{n+k-1}{k} = \frac{(n+k-1)!}{k!(n-1)!},$$
where in this case $k=2$ and $n=N$, so
$$ E = \frac{(n+1)!}{2!(n-1)!} = \frac{(n+1)n}{2}$$