TIFR CSE 2023 | Part A | Question: 11

Question

We are given a set $S=\left\{x_{1}, \ldots, x_{n}\right\}$ of distinct positive integers such that $\operatorname{gcd}\left(x_{i}, x_{j}\right)= 1$ for any $i, j \in\{1, \ldots, n\}$ where $i \neq j$. What is the total number of invertible $2 \times 2$ matrices whose entries are distinct elements from the set $S$ ?

$\textbf{(Note:}$ For positive integers $a$ and $b, \operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b)$.

• A. $n^{4}$
• B. $(n-1)^{4}$
• C. $n^{2}(n-1)^{2} / 4$
• D. $n(n-1)(n-2)(n-3)$
• E. $n(n-1)(n-2)(n-3) / 4$ !

   

Answer 1

ans: D.

GCD(Xi,Xj)=1 means they are prime to each other(coprime).so all theelements are coprime to each other.therefore for 2*2 matrices we have all distinct elements in 4 positions  so its like picking 4 elements from n element set nC4 and further they can arrange among themselves=nC4*4!.=n(n-1)(n-2)(n-3).