1. **Problem:** Find the value of $n$ if the number of terms in the expansion of $(1 - 4x + 4x^2)^n$ is 7. 2. **Formula:** The number of terms in the expansion of a trinomial $(a + b + c)^n$ is given by the formula for combinations with repetition: $$\frac{(n+1)(n+2)}{2}$$ 3. **Step:** Set the number of terms equal to 7: $$\frac{(n+1)(n+2)}{2} = 7$$ 4. **Simplify:** Multiply both sides by 2: $$(n+1)(n+2) = 14$$ 5. **Expand:** $$n^2 + 3n + 2 = 14$$ 6. **Rearrange:** $$n^2 + 3n + 2 - 14 = 0$$ $$n^2 + 3n - 12 = 0$$ 7. **Solve quadratic:** Using the quadratic formula: $$n = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-12)}}{2} = \frac{-3 \pm \sqrt{9 + 48}}{2} = \frac{-3 \pm \sqrt{57}}{2}$$ 8. **Approximate:** Since $n$ must be a non-negative integer, approximate $\sqrt{57} \approx 7.55$: $$n = \frac{-3 + 7.55}{2} = 2.275$$ (not integer) 9. **Check integer values:** Try $n=3$: $\frac{(3+1)(3+2)}{2} = \frac{4 \times 5}{2} = 10$ (too big) Try $n=2$: $\frac{(2+1)(2+2)}{2} = \frac{3 \times 4}{2} = 6$ (too small) Try $n=4$: $\frac{(4+1)(4+2)}{2} = \frac{5 \times 6}{2} = 15$ (too big) 10. **Conclusion:** The problem states the number of terms is 7, which is not possible for integer $n$ using the formula. However, the closest integer $n$ that yields a number of terms near 7 is $n=3$ (10 terms) or $n=2$ (6 terms). Since the problem options include 3, 4, 6, 8, the best fit is $n=3$. **Final answer:** $n = 3$ (Option ক) --- **Slug:** trinomial_terms **Subject:** algebra **desmos:** {"latex":"y= (1 - 4x + 4x^2)^n","features":{"intercepts":true,"extrema":true}} **q_count:** 13