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One-One Functions with Constraints: 72 | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question
Let A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\}. The number of one-one functions f:A→Af:A\to A such that f(1)≥3f(1)\ge3, f(3)≤4f(3)\le4 and f(2)+f(3)=5f(2)+f(3)=5, is
Solution
Answer: 72 (± 0.01)
Step 1: Distinct pairs (f(2),f(3))(f(2),f(3)) with f(2)+f(3)=5f(2)+f(3)=5, f(3)≤4f(3)\le4: (1,4),(4,1),(2,3),(3,2)(1,4),(4,1),(2,3),(3,2) all qualify. Step 2: Take f(2)=4, f(3)=1f(2)=4,\ f(3)=1. f(1)≥3f(1)\ge3 and distinct from {4,1}\{4,1\}: f(1)∈{3,5,6}f(1)\in\{3,5,6\}, 33 choices. Remaining 33 values fill f(4),f(5),f(6)f(4),f(5),f(6) in 3!=63!=6 ways: 3×6=183\times6=18. Step 3: Each of the 44 pairs leaves exactly 33 choices for f(1)f(1), so 1818 each:
Total=4×18=72.\text{Total}=4\times18=72.
 Correct answer: 72
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