Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
The total number of different ways a grandfather along with two of his grandsons and four grand daughters can be seated in a line for a photograph so that he is always in the middle and the two grandsons are never adjacent to each other:
Step-by-Step Solution
Key Concept: Substituting specific values and using the functional equation $f(-x)$ reveals coefficient relationships through comparison.
For $f(x) = (1 + x^2 - x^3)^{1000} = a_0 + a_1x + ... + a_{3020}x^{3020} + ...$, we substitute $x = 1$ and $x = -1$ to establish relationships. Since $f(-x) = (1 + x^2 + x^3)^{1000} = a_0 - a_1x + ... + a_{3020}x^{3020} + ...$, comparing coefficients shows that even-indexed coefficients are symmetric: $a = d$ and $b = c$, making $d$ the largest coefficient.
Correct Answer: 3