Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
How many five digit numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the unit place must be greater than that in the tenth place?
54
60
$51/3$
$2 \times 4!$
Step-by-Step Solution
Key Concept: Factorization combined with coefficient extraction requires separating the expansion into manageable polynomial forms and analyzing divisibility conditions.
We expand $(x^3-1)^n = (x-1)^n(1+x+x^2)^n$ and find the coefficient of $x^n$ in the product. This equals $\sum_{r=0}^{n}C_r(-1)^r a_r$ where $a_r$ is the coefficient in $(x^3-1)^n$. The result depends on whether $n$ is divisible by 3: if $3|n$, the coefficient is $^nC_{2n/3}(-1)^{n/3}$ and $^nC_n$, otherwise it is 0.
Correct Answer: 2