If $a * b = 2a - 3b + ab$, then $3 * 5 + 5 * 3$ is equal to

### Concept & Logic This problem requires evaluating a non-commutative binary operation for two different pairs of inputs and adding their results. Because the operation is not commutative ($a * b \neq b * a$), you must calculate each term separately. ### Step-by-Step Solution * **Given:** $a * b = 2a - 3b + ab$ * **Step 1: Calculate $3 * 5$** Substitute $a = 3$ and $b = 5$: $$3 * 5 = 2(3) - 3(5) + (3)(5)$$ $$3 * 5 = 6 - 15 + 15$$ $$3 * 5 = 6$$ * **Step 2: Calculate $5 * 3$** Substitute $a = 5$ and $b = 3$: $$5 * 3 = 2(5) - 3(3) + (5)(3)$$ $$5 * 3 = 10 - 9 + 15$$ $$5 * 3 = 16$$ * **Step 3: Add the results** $$(3 * 5) + (5 * 3) = 6 + 16 = 22$$ ### Exam Strategy & Shortcut Notice that the $+ab$ term is commutative and will be the same for both $3 * 5$ and $5 * 3$. You can group the expression algebraically before computing fully: $(2a_1 - 3b_1 + a_1b_1) + (2a_2 - 3b_2 + a_2b_2)$ where the pairs are reversed. Sum = $2(3+5) - 3(5+3) + 2(15) = 2(8) - 3(8) + 30 = 16 - 24 + 30 = 22$. This algebraic grouping can sometimes be faster than computing two separate arithmetic chains. ### Common Pitfall Assuming the operation is commutative ($3 * 5 = 5 * 3$) and just calculating one side and doubling it. Always check if the formula is symmetrical. Here, $2a - 3b$ is asymmetrical, so the terms must be calculated independently. ### Final Answer **Therefore, the correct answer is 22.**