JEE MAIN - Mathematics (2024 - 30th January Morning Shift - No. 29)
Let $$\mathrm{A}=\{1,2,3, \ldots, 7\}$$ and let $$\mathrm{P}(\mathrm{A})$$ denote the power set of $$\mathrm{A}$$. If the number of functions $$f: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})$$ such that $$\mathrm{a} \in f(\mathrm{a}), \forall \mathrm{a} \in \mathrm{A}$$ is $$\mathrm{m}^{\mathrm{n}}, \mathrm{m}$$ and $$\mathrm{n} \in \mathrm{N}$$ and $$\mathrm{m}$$ is least, then $$\mathrm{m}+\mathrm{n}$$ is equal to _________.
Answer
44
Explanation
$$\begin{aligned} & f: A \rightarrow P(A) \\ & a \in f(a) \end{aligned}$$
That means '$$a$$' will connect with subset which contain element '$$a$$'.
Total options for 1 will be $$2^6$$. (Because $$2^6$$ subsets contains 1)
Similarly, for every other element
Hence, total is $$2^6 \times 2^6 \times 2^6 \times 2^6 \times 2^6 \times 2^6 \times 2^6=2^{42}$$
Ans. $$2+42=44$$
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