Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
The total number of integral solution for $x, y, z$ such that $xyz=24$ is:
Step-by-Step Solution
Key Concept: Coefficient extraction from binomial expansions using $(-1)^r C_r^n$ patterns requires recognizing the underlying polynomial structure.
We need to find the coefficient of $x^n$ in $-\sum_{r=1}^{n}(-1)^r C_r^n = -n\sum_{r=1}^{n}(-1)^{n-1}C_{r-1}^n C_r$. Using binomial expansion properties, we recognize this as finding the coefficient of $x^n$ in $(1-x^2)^{n+1}(1-x)$. The alternating sum with binomial coefficients yields the desired coefficient through careful application of the binomial theorem and coefficient matching.
Correct Answer: 4