Let $A=\{1,2,3,\ldots,7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f:A\rightarrow P(A)$ such that $a\in f(a)$, $\forall a\in A$ is $m^{n}$, $m$ and $n\in\mathbb{N}$ and $m$ is least, then $m+n$ is equal to

JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Sets Relations and Functions

Let

A = {1, 2, 3, \dots, 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

m^{n}

m

and

n \in N

and

m

is least, then

m + n

is equal to

Answer: 44

Each element

a \in A

needs

f (a)

to be a subset of

A

containing

a

; there are

2^{6}

such subsets (the other 6 elements each independently in or out).

This choice is made independently for each of the 7 elements of

A

, so the total number of functions is

(2^{6})^{7} = 2^{42}

.

With

m^{n} = 2^{42}

and

m

least,

m = 2, n = 42

, so

m + n = 44

JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Integer answer