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JEE Main 2026 · 5 Apr · Shift 1, Mathematics question 21

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 ________.

Answer and worked solution

Official answer: 72

Pairs (f(2),f(3))(f(2),f(3)) with sum 5 and f(3)≤4f(3)\le4: (1,4),(2,3),(3,2),(4,1)(1,4),(2,3),(3,2),(4,1).

Each pair. f(1)≥3f(1)\ge3 and must differ from both values:

  • (1,4)(1,4) and (4,1)(4,1): f(1)∈{3,5,6}f(1)\in\{3,5,6\}.
  • (2,3)(2,3) and (3,2)(3,2): f(1)∈{4,5,6}f(1)\in\{4,5,6\}.

So there are 3 choices each time, and the remaining 3 values fill f(4),f(5),f(6)f(4),f(5),f(6) in 3!=63!=6 ways.

Total: 4⋅3⋅6=724\cdot3\cdot6=72.

Answer: 72

Topics:Fundamental principle of countingTypes of functions
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