Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
In an election three districts are to be canvassed by 2, 3 and 5 men respectively. If 10 men volunteer, the number of ways they can be allotted to the different districts is:
10!/(2!*3!*5!)
10!/(2!*5!)
10!/((2!)^3*5!)
10!/(2!*2!*2!*5!)
Step-by-Step Solution
Key Concept: The binomial expansion of conjugate surds separates into integer and irrational parts, with floor function constraints yielding unique solutions.
For the equation $(5\sqrt{3} + 8)^{2n+1} - (5\sqrt{3} - 8)^{2n+1} = 2({}^{2n+1}C_1(5\sqrt{3})^{2n}8 + {}^{2n+1}C_3(5\sqrt{3})^{2n-2}8^3 + ...)$, if $\lfloor x \rfloor + \lfloor x \rfloor - N = 2k$ where $k \in \mathbb{Z}$ and $0 < \lfloor x \rfloor < 1$, then $-1 < \lfloor x \rfloor - N < 1$ with $\lfloor x \rfloor - N \in \mathbb{Z}$ implies $\lfloor x \rfloor - N = 0$, so $2N = \lfloor x \rfloor = (11)^{2n+1}$.
Correct Answer: 1