Let A=1,2,3, ,7 and let P(A) denote the power set of A.

Question

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

Answer: 44

Accepted Answer

$\begin{gathered} f:A	o P(A) \ a\in f(a) \end{gathered}$

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 ${2}^{6}	imes {2}^{6}	imes {2}^{6}	imes {2}^{6}	imes {2}^{6}	imes {2}^{6}	imes {2}^{6}={2}^{42}$

Ans. $2+42=44$

Solutions · 1