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JEE Main20238 Apr 2023Evening ShiftMathematicsFunctionsActual

Let R = a , b , c , d , e and S = 1 , 2 , 3 , 4 . Total number of onto functions f : R → S such that f ( a ) ≠ 1 , is equal to ________.

Correct answer

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Step-by-step solution

Given, R = a , b , c , d , e and S = 1 , 2 , 3 , 4 Now taking, f ( a ) = 1 we get, one of f ( b ) ,   f ( c ) ,   f ( d ) ,   f ( e ) = 1 then total such cases = 4 · 3 ! = 24 Now if, only f ( a ) = 1 , then we have distribute 2 , 3 , 4 amongst b , c , d , e , So, total cases = 3 4 - C 1 3 · 2 4 + C 2 3 · 1 = 36 So, number of onto functions when f ( a ) = 1 is 24 + 36 = 60 Now finding, total number of onto functions, = 4 5 - C 1 4 · 3 5 + C 2 4 · 2 5 - C 3 4 · 1 = 1024 -

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