Permutations & Combinations
Permutations & Combinations
star_batch_jee_advanced_2025
Grade 11
Question:
Three digit numbers in which the middle one is a perfect square are formed using the digits $1$ to $9$. Their sum is:
134055
270540
170055
None of these
Step-by-Step Solution
Key Concept: Decompose higher-order terms using $r^2 = r(r-1) + r$ and apply standard binomial coefficient reduction identities.
Starting with $n\sum_{r=1}^{n} r^2 ^{n-1}C_{r-1}p^rq^{n-r}$, expand using $r^2 = r(r-1) + r$ to split into two sums. Apply the identity $r(r-1)\binom{n-1}{r-1} = n(n-1)\binom{n-2}{r-2}$ and simplify each component systematically to arrive at the final result $np(n-1)(n-2)p^2 + 3n(n-1)p^2 + np = np[(n-1)(n-2)p^2 + 3(n-1)p + 1]$.
Correct Answer: 1