Question

Let S = {1, 2, 3, 4, 5}
if f : S → P(S), where P(S) is power set of S. Then number of one-one functions f can be made is

• A. (32)⁵
• B. \(\frac{32!}{27!}\)
• C. \(^{32}C_{27}\)
• D. \(^{32}P_{27}\)

Answer 1

The Correct Option is B

The given problem involves finding the number of one-one functions from a set \( S \) to its power set \( P(S) \). Let us go through this step-by-step:

Understanding the Problem:

• Set \( S \) is given as \( \{1, 2, 3, 4, 5\} \) which has 5 elements.
• The power set \( P(S) \) is the set of all subsets of \( S \). The number of subsets of a set with \( n \) elements is \( 2^n \). Therefore, \( P(S) \) has \( 2^5 = 32 \) elements.

Key Concept: One-One Function

• A one-one (injective) function from a set \( A \) to a set \( B \) assigns distinct elements of \( B \) to each element of \( A \).
• The number of one-one functions from a set with \( m \) elements to a set with \( n \) elements (where \( n \geq m \)) is calculated by choosing \( m \) elements from \( n \) and arranging them, which is the permutation \( ^nP_m = \frac{n!}{(n-m)!} \).

Application to the Problem:

• Here, \( m = 5 \) (elements in \( S \)) and \( n = 32 \) (elements in \( P(S) \)).
• We need to find the number of one-one functions from \( S \) to \( P(S) \), which is \( ^{32}P_5 \).
• The formula for permutation is given by \( \frac{32!}{(32-5)!} = \frac{32!}{27!} \).

Conclusion:

• The correct answer is \(\frac{32!}{27!}\). This matches option 2 provided in the question.