Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
Six persons $A, B, C, D, E$ and $F$ are to be seated at a circular table. The number of ways this can be done if $A$ must have either $B$ or $C$ on his right and $B$ must have either $C$ or $D$ on his right is:
Step-by-Step Solution
Key Concept: Recognize and apply identities like $r\binom{n}{r} = n\binom{n-1}{r-1}$ combined with convolution formulas for binomial coefficients.
Starting with $\sum_{r=1}^{n}(r^nC_r)[(n-r)^nC_r] = n\sum_{r=1}^{n} ^{n-1}C_{r-1}(n-r)^nC_r$, apply the identity $(n-r)^nC_r = n^{n-1}C_{r-1}$ to simplify. Use binomial convolution and properties of repeated indices to show this equals $n^2\sum_{r=1}^{n} ^{n-2}C_{r-2} ^{n-2}C_{r-1} = n(n-1)2^{n-2}C_{n-1}$.
Correct Answer: 4