Number of Onto Functions from \(A=\{1,2,\dots,n\}\) to Itself

📺 Video Explanation

📝 Question

Find the number of all onto functions:

\[ f:A\to A \]

where:

\[ A=\{1,2,3,\dots,n\} \]

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✅ Solution

Set \(A\) has:

\[ n \]

elements.

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🔹 Key Fact

For finite sets with same number of elements:

\[ \text{onto} \iff \text{one-one} \]

So every onto function from \(A\) to itself is a permutation.

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🔹 Number of Permutations

Number of ways to arrange:

\[ n \]

distinct elements:

\[ n! \]

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🎯 Final Answer

\[ \boxed{n!} \]

Total number of onto functions from \(A\) to itself is:

\[ \boxed{n!} \]

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🚀 Exam Shortcut

• Same size finite sets:
• Onto = one-one = permutation
• Number of permutations = \(n!\)