The associative property of addition states that the way in which numbers are grouped in an addition problem does not change the sum. Mathematically, it can be stated as:
[tex]\[
(a + b) + c = a + (b + c)
\][/tex]
To determine which expression illustrates this property, let's examine each option one-by-one:
1. [tex]\((3 + 19) - 12 = (3 + 12) - 19\)[/tex]
Rearranging terms in this expression does not follow the associative property of addition because subtraction is involved and the grouping is altered in terms of both addition and subtraction. This is not an example of the associative property.
2. [tex]\(3 + (19 - 12) = 3 + (19 + 12)\)[/tex]
This expression shows a sum on one side and a different sum on the other, which changes the operation inside the parentheses rather than just regrouping the numbers. This does not follow the associative property of addition.
3. [tex]\((3 + 19) - 12 = 3 + (19 - 12)\)[/tex]
This arrangement shows a regrouping of numbers within addition and subtraction but does not follow the associative property exactly as defined. The right-hand side adjusts subtraction within the addition context but is not a pure grouping change.
4. [tex]\(3 + (19 - 12) = 3 - (19 + 12)\)[/tex]
Here, the operations differ on each side (addition versus subtraction), and this does not exemplify the associative property of addition since no pure regrouping is taking place.
Upon careful inspection of all the expressions, none of them conform directly to the standard [tex]\(a + (b + c) = (a + b) + c\)[/tex]. However, if we evaluate these expressions, the one that correctly maintains the relationship implied by simplification balances is:
[tex]\[
(3 + 19) - 12 = 3 + (19 - 12)
\][/tex]
This relationship ultimately holds true, illustrating that despite the complexity and non-clear demonstration of associative property directly, when evaluated, it satisfies the specific numeric equality.
Therefore, the correct expression that best aligns with the context of associative properties of addition (though not perfect) is:
[tex]\[
(3 + 19) - 12 = 3 + (19 - 12)
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{3}
\][/tex]
