JEE Mathematics 2024 Question with Solution

Given: $A = \{1,2,3,4,5,6,7\}$ and $f: A \to P(A)$ with the condition $a \in f(a)$ for all $a \in A$.

Find: The value of $m+n$ if the number of such functions is $m^n$ with least possible $m$.

For each fixed element $a \in A$, the subset $f(a)$ must contain $a$.

A subset of $A$ containing $a$ is formed by keeping $a$ fixed and choosing independently whether each of the remaining $6$ elements is included or not.

So, the number of possible values of $f(a)$ is

$$2^6$$

Since this choice is independent for each of the $7$ elements of $A$, the total number of such functions is

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

Hence,

$$m^n = 2^{42}$$

To make $m$ least, take the smallest base possible, namely $m=2$. Then

$$2^n = 2^{42} \implies n=42$$

Therefore,

$$m+n = 2+42 = 44$$

So, the required numerical value is $44$.

Given: For every $a \in A$, the image $f(a)$ must be a subset of $A$ containing $a$.

Find: $m+n$.

Think elementwise: for each input element, one element is already forced to be present in its image subset, so only the other $6$ elements are free.

Thus each input has

$$2^6$$

choices, and with $7$ independent inputs the total number of functions is

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

Therefore the least-base representation is $m=2$ and $n=42$.

So, the answer is $44$.