JEE Main 2022 · 28 Jun · Shift 2 · Mathematics Q29

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MathematicsInteger+4 / -1Question 29 of 90

Let $S=\{1,2,3,4\}$ . Then the number of elements in the set $\{f:S\times S\to S:f\text{ is onto and }f(a,b)=f(b,a)\ge a\ \forall(a,b)\in S\times S\}$ is ______.

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Answer and worked solution

Official answer: 37

Condition. $f(a,b)\ge\max(a,b)$ on each unordered pair. Value 1 is possible only at $(1,1)$ , so $f(1,1)=1$ ; pairs with max 4 must be 4.

Pairs with max 2 (two of them) take values in $\{2,3,4\}$ with at least one 2: 5 ways. Max 3 (three pairs): $\{3,4\}$ , 8 ways.

Need a 3 somewhere. $5\cdot8-3\cdot1=37$ .

Answer: 37

Topics:Types of functions Fundamental principle of counting

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JEE Main 2022 · 28 Jun · Shift 2, Mathematics question 29