Question

Give a big-o estimate for the number of operations (where an operation is an addition or a multiplication) used in this segment of an algorithm. t := 0 for i := 1to3 for j := 1to4 t := t + ij

Answer 1

We have a "rectangular" double loop, meaning that both loops go to completion.
So there are 3*4=12 executions of t:=t+ij.

Assuming two operatiions per execution of the innermost loop, (i.e. ignoring the implied additions in increment of subscripts), we have 12*2=24 operations in all.

Here the number of operations (+ or *) is exactly known (=24). 

Big-O estimates are used for cases with a varying scale of operations, governed by a variable (usually n) to indicate the sensitivity of the number of operations relative to a change in the size of n.

Here we do not have a scale, nor n is defined.  The number of operations is constant and known at 24.   So a variable is required to find the big-O estimate.