Let A = 1,2,3,4,5,6,7 . The number of surjective functions defined from A to A such that f ( i )= i for atleast four values of i from i =1,2, ldots . ., 7, is

Question

Let A = 1,2,3,4,5,6,7 . The number of surjective functions defined from A to A such that f ( i )= i for atleast four values of i from i =1,2, ldots . ., 7, is (A) 7 ! (B) 92 (C) 126 (D) 407

Tardigrade Admin · Accepted Answer

A= 1,2,3,4,5,6,7 Case - I When exactly 4 values follows f ( i )= i 7 C 4 × 3 !((1/2 !)-(1/3 !))=70 Case - II When exactly 5 values follows f ( i )= i 7 C 5 × 1=21 Case - III When all 7 values follows f ( i )= i, number of function =1 Total functions =70+21+1=92

Q. Let A={1,2,3,4,5,6,7}. The number of surjective functions defined from A to A such that f(i)=i for atleast four values of i from i=1,2,…..,7, is

7 !

92

126

407

A={1,2,3,4,5,6,7} Case - I When exactly 4 values follows f(i)=i 7C4​×3!(2!1​−3!1​)=70 Case - II When exactly 5 values follows f(i)=i 7C5​×1=21 Case - III When all 7 values follows f(i)=i, number of function =1 Total functions =70+21+1=92

Q. Let $A = {1, 2, 3, 4, 5, 6, 7}$ . The number of surjective functions defined from $A$ to $A$ such that $f (i) = i$ for atleast four values of $i$ from $i = 1, 2, \dots .., 7$ , is