Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Question:
Number of ways in which 7 green bottles and 8 blue tube bottles can be arranged in a row if exactly 1 pair of green bottles is side by side, is (Assume all bottles to be alike except for the colour):
Step-by-Step Solution
Key Concept: Separating binomial expansions into even and odd indexed terms creates elegant identities relating sums and products of these partial sums.
Given $\alpha = ^nC_0 x^n + ^nC_2 x^{n-2}a^2 + ^nC_4 x^{n-4}a^4 + \ldots$ and $\beta = ^nC_1 x^{n-1}a + ^nC_3 x^{n-3}a^3 + ^nC_5 x^{n-5}a^5 + \ldots$, we recognize these as the even and odd binomial terms from $(x+a)^n$ and $(x-a)^n$. Using $(x+a)^{2n} - (x-a)^{2n} = 4a\alpha\beta$, $(x+a)^{2n} + (x-a)^{2n} = 2(\alpha^2 + \beta^2)$, and $(x+a)^n(x-a)^n = \alpha^2 - \beta^2$.
Correct Answer: 3