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+1 vote
How many one-to-one functions are there from a set $A$ with $n$ elements onto itself?
asked in Set Theory & Algebra by Veteran (39.8k points) 258 1313 1945 | 197 views

2 Answers

+5 votes
Best answer
There are n! one to one function possible with n elements to itself.. I.e. $P\binom{n}{n}$ = n!
answered by Veteran (53k points) 21 72 327
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All these n! functions will also be onto and so they all are bijections ...
0 votes

f : A---> A

∣A∣ = n

The first element of the domain has n choices for mapping,2nd element has (n-1) ,3rd element has (n-2) choices and so on.

So, total number of one-to-one functions = n ⨉(n-1)⨉(n-2)⨉(n-3).........⨉1 = n!

The correct answer is n! .

answered by Boss (9.4k points) 4 8 13

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