Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Question:
The number 916238457 is an example of nine digit number which contains each of the digit 1 to 9 exactly once. It also has the property that the digits 1 to 5 occur in their natural order, while the digits 1 to 6 do not. Number of such numbers are:
Step-by-Step Solution
Key Concept: Use coefficient recurrence relations and summation identities to transform products of polynomials and binomial terms.
Express $\sum_{r=0}^{n} \frac{r^2}{^nC_r} = a_n + \sum_{r=0}^{n} \frac{(r-1)(r+1)}{^nC_r}$. Use the recurrence relation for binomial coefficients and telescoping to obtain $a_n + (n+1)(n+2)\sum_{r=0}^{n} \frac{1}{^{n+2}C_{r+2}} - 3(n+1)\sum_{r=0}^{n} \frac{1}{^{n+1}C_{r+1}}$. Evaluate the resulting sums using the identity $a_n = (n+1)(n+2)a_{n-2}$ to find the final closed form.
Correct Answer: 2