Conditional Function Mapping

Total number of functions possible = 3125 As for each element in domain 5 choices are there, so total functions 5^5 Total number of onto function possible=120 As for first element there is 5 options , for second there is 4 options and so on. So total onto functions possible 5! hence total number of into functions 3125-120=3005

Now let us count total functions where f(i) $\neq$ i For i=1 we have 4 options {2,3,4,5}, similarly we have 4 options for i=2{1,3,4,5} Hence total functions where f(i) $\neq$ i=4^5=1024 However, out of these not all are into functions Let us count total onto functions where f(i) $\neq$ i Since 1 should not be mapped with 1 , 2 should not be mapped with 2 and so on; further each of {1,2,3,45} should be uniquely mapped so number of onto functions where f(i) $\neq$ i = Derangement of 5 =44 Hence, total onto functions where f(i) $\neq$ i = 44 Therefore, total into functions where f(i) $\neq$ i=1024-44 Hence, total into functions where f(i)=i for atleast one of i={1,2,3,4,5} =3005-(1024-44) = 2025 Hence required probability= $\frac { 2025 }{ 3005 }$ = $\frac { 405 }{ 601 }$