Question

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

• A. 92
• B. 92.0
• C. 92.00

Answer 1

Answer: (A), (B), (C)

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