Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it, so that there will be no complete pair is:
Step-by-Step Solution
Key Concept: Symmetric coefficient sequences ($a_r = a_{2n-r}$) allow splitting sums into equal parts and using total sum constraints to find partial sums.
Given that $a_r = a_{2n-r}$ and $\sum_{r=0}^{2n}a_r = 3^n$, we exploit symmetry to compute $a_0 + a_1 + \ldots + a_{n-1} = \frac{3^n - a_n}{2}$. From the expansion $(1-x+x^2)^n = a_0x^{2n} - a_1x^{2n-1} + a_2x^{2n-2} - \ldots$, extracting coefficients and using symmetry conditions yields the desired coefficient relationships.
Correct Answer: 4