1. Problem statement: We are given $a^3 + b^3 = 217$ and $a + b = 7$. 2. Goal: Find $ab$. 3. Formula and rule: Use the identity $$a^3 + b^3 = (a+b)^3 - 3ab(a+b)$$ This identity follows from expanding $$(a+b)^3 = a^3 + b^3 + 3ab(a+b)$$ 4. Substitute the known values into the identity. $$217 = 7^3 - 21ab$$ 5. Solve for $ab$ by rearranging. $$21ab = 7^3 - 217 = 343 - 217 = 126$$ $$ab = 126/21 = 6$$ 6. Final answer: Therefore $ab = 6$. The correct choice is (b) 6.