Question

Function $f:\{1,2,3,4,5\}\to \{1,2,3,4,5\}$ is said to be nasty if there do not exist distinct $a,b\in \{1,2,3,4,5\}$ satisfying $f\left(a\right)=b$ and $f\left(b\right)=a.$ How many nasty functions are there?

• A. $1950$
• B. $2025$
• C. $3125$
• D. $1250$

Answer 1

The correct option is A $1950$

Total number of functions $=5^5=3125$
Number of functions whose two elements mapping to each other $={}^5{C}_2\cdot 5^3=1250$
Some pairs are counted twice like $\left\{f\right(1)=2,f(2)=1\},\left\{f\right(1)=2,f(2)=1,f(3)=4,f(4)=3\}$
Number of pairs which are over count $={}^5{C}_4\cdot 3\cdot 5^1=75$
Required number of functions $=3125-(1250-75)=1950$