Question

One mapping (function) is selected at random from all the mappings of the set \(A = \{1, 2, 3, \dots, n\}\) into itself. The probability that the mapping selected is one-one, is

• A. \(\frac{n!}{n^{n-1}}\)
• B. \(\frac{n!}{n^n}\)
• C. \(\frac{n!}{2n^n}\)
• D. None of these

Hint:

For one-one mapping, each element in codomain gets at most one preimage.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Total mappings = \(n^n\), one-one mappings = \(n!\).
Step 2: Detailed Explanation:
Total number of functions from \(A\) to \(A\) = \(n^n\)
Number of injective (one-one) functions = \(n!\)
Probability = \(\frac{n!}{n^n}\)
Step 3: Final Answer:
\(\frac{n!}{n^n}\).