1. Tardigrade
  2. Question
  3. Mathematics
  4. 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

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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 $i =1,2, \ldots . ., 7$, is

Relations and Functions - Part 2

Solution:

$ A=\{1,2,3,4,5,6,7\}$
Case - I When exactly 4 values follows $f ( i )= i$
${ }^7 C _4 \times 3 !\left(\frac{1}{2 !}-\frac{1}{3 !}\right)=70$
Case - II When exactly 5 values follows $f ( i )= i$
${ }^7 C _5 \times 1=21$
Case - III When all 7 values follows $f ( i )= i$,
number of function $=1$
Total functions $=70+21+1=92$