NCERT Class 12 · Ch 1 Miscellaneous Exercise · Q4
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Miscellaneous Exercise Question 4
Find the number of onto functions from {1,2,…,n} to itself, for n≥1.
💡 Concept Hints
- Onto between two finite sets of the same size forces every output to be used exactly once.
- Write the defining condition carefully and preserve every domain restriction while simplifying.
✅ Step-by-Step Solution
Final Answer: n!.
- Both finite sets have n elements. If an onto function repeated a value, its range would contain at most n−1 elements, a contradiction.
- Thus an onto self-map must be a bijection. Its images form a permutation of the n elements.
- Choose the first image in n ways, the second in n−1 ways, and continue to the last in 1 way.
- The total is n(n−1)⋯1=n!.
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