JEE Main 2024 (Online) 30th January Morning ShiftQuestion 1
Let $A=\{1,2,3,\ldots,7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f:A\to P(A)$ such that $a\in f(a)$, $\forall a\in A$ is $m^{n}$, $m$ and $n\in\mathbf{N}$ and $m$ is least, then $m+n$ is equal to $\_\_\_\_\_\_$ .
Sets, Relations and Function
Answer$44$
Solution
Each $f(a)$ is a subset containing $a$: other 6 elements free, $2^6$ choices. Number of functions $=(2^6)^7=2^{42}=m^n$. Least $m=2$, $n=42$. $m+n=44$.