Question

Consider the function $f:A\to A$ where $A:\left\{1,2,3,4,5\right\}$ which satisfy the conditions $f\left(f\left(x\right)\right)=x$. if the number of such functions are $\lambda$, then

• A. Sum of digits of $\lambda =8$
• B. No of prime factors of $\lambda =41$
• C. Product of digits of $\lambda =2$
• D. Product of digits of $\lambda =12$

Answer 1

The correct option is B No of prime factors of $\lambda =41$

$f:A\to A$

$A=\{1,2,3,4,5\}$

$f\left(f\right(x\left)\right)=x$

that means

$f\left(x\right)=f^{-1}\left(x\right)$

Total function $5!=120$

which are invertible

$\lambda ={}^5{C}_2\times {}^3{C}_2+{}^5{C}_2+1=41$

$\downarrow$ $\downarrow$ $\downarrow$

select a select a select a

pair from pair from pair from

$5$ $5$ $5$

Pair two numbers $a$ & $b$ such that $f\left(a\right)=bf\left(b\right)=a$

then $f\left(f\right(a\left)\right)=af\left(f\right(b\left)\right)=b$