JEE Mathematics 2025 Question with Solution

Step-by-step solution

Common mistakes

• Counting ordered triples instead of selected sets is incorrect because the question asks for three numbers to be selected, not arranged. Use combinations $\binom{40}{3}$ for the total sample space.

• Assuming only integer common ratios misses valid G.P. triplets such as those with ratio $\frac{3}{2}$. For integer terms, use the form $\left(kq^2,kqp,kp^2\right)$ with coprime integers $p>q\ge 1$.

• Forgetting the condition that all three terms must lie in $\{1,2,\dots,40\}$ leads to overcounting. Always impose the bound on the largest term, namely $kp^2 \le 40$.