Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
Triplet $(x, y, z)$ is chosen from the set $\{1, 2, 3, ..., n\}$, such that $x \leq y < z$. The number of such triplets is:
n^3
^{n+1}C_3
^nC_2
^nC_2 + ^nC_3
Step-by-Step Solution
Key Concept: Alternating binomial sums often telescope or simplify via factorization when viewed through generating functions.
The coefficient of $x^n$ in $\binom{n}{0}(1+x)^{2n} - \binom{n}{1}(1+x)^{2n-1} + \binom{n}{2}(1+x)^{2n-2} - \ldots + (-1)^n\binom{n}{n}(1+x)^0$ equals the coefficient of $x^n$ in $(1+x)^n((1+x)-1)^n = (1+x)^n \cdot x^n$, which is 1.
Correct Answer: 2,4