Question

Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \rightarrow P(A) \) 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 \(\_\_\_\_\_\_\_\_\_\).

Answer 1

Correct Answer: 44

Solution: Each element \( a \) in \( A \) must be included in its corresponding subset in \( P(A) \), so we only consider subsets of \( A \) that contain \( a \). For each element, there are \( 2^6 \) possible subsets of \( A \) that include \( a \) (since we can select or omit any of the remaining 6 elements).

Thus, for each \( a \in A \), there are \( 2^6 \) choices, and since there are 7 elements in \( A \):

Total number of functions = \( (2^6)^7 = 2^{42} \)

Since we need \( m^n = 2^{42} \) with \( m \) and \( n \) as small as possible:

\( m = 2 \), \( n = 42 \)

Therefore, \( m + n = 2 + 42 = 44 \).