Question

Let `f:A->A` be an invertible function where `A= {1,2,3,4,5,6}` The number of these functions in which at least three elements have self image is

Answer 1

Correct Answer - 56
If exactly r element have self image then number of functions =(Number of ways of selecting r elements)`xx` (Derangement of remaining elements)
`=""^(n)C_(r ) r!(1-(1)/(1!)+(1)/(2!)-..)`
`therefore` Required number of functions
`= ""^(6)C_(3)xx3!xx(1-(1)/(1!)+(1)/(2!)-(1)/(3!))+ ""^(6)C_(4)xx2!(1-(1)/(1!)+(1)/(2!))+""^(6)C_(5)xx1!xx(1-(1)/(1!))+""^(6)C_(6)`
=40+15+0+1
=56