1. **State the problem:** Given that $a^3 + b^3 = 217$ and $a + b = 7$, find the value of $ab$. 2. **Recall the formula:** The sum of cubes can be expressed as $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ 3. **Use the given values:** Substitute $a + b = 7$ into the formula: $$217 = 7(a^2 - ab + b^2)$$ 4. **Express $a^2 - ab + b^2$ in terms of $a+b$ and $ab$:** We know that $$a^2 + b^2 = (a + b)^2 - 2ab = 7^2 - 2ab = 49 - 2ab$$ Therefore, $$a^2 - ab + b^2 = (a^2 + b^2) - ab = (49 - 2ab) - ab = 49 - 3ab$$ 5. **Substitute back into the equation:** $$217 = 7(49 - 3ab)$$ 6. **Simplify:** $$217 = 343 - 21ab$$ 7. **Isolate $ab$:** $$217 - 343 = -21ab$$ $$-126 = -21ab$$ 8. **Divide both sides by $-21$:** $$\cancel{-21}ab = \cancel{-21} \frac{-126}{-21}$$ $$ab = 6$$ **Final answer:** $ab = 6$ which corresponds to option (b).