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Is there a set formula to find No. Of one-one function? If so, what theory is involved with it?

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    Combinatorics.. – 2017-01-20
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    Permutation and combinations? – 2017-01-20
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    Could u pls tell me the reason why it is being used? – 2017-01-20

4 Answers 4

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Let $A$ be a set of $n$ elements, and $B$ be a set of $m$ elements, $n,m \in \mathbb{Z}$. To give a $1-1$ function $f : A \to B$ is the same as to say what $Im(f) \subseteq B$ should be, and then give a bijection $A \to Im(f)$. There are $m \choose n$ ways to decide $Im(f)$, and $n!$ ways to give the bijection.

Hence the answer is ${m \choose n} n!$.

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    Im(f) is image of f? – 2017-01-20
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    Yes! It is those elements in $b \in B$ such that $f(a) = b$ for some $a \in A$ – 2017-01-20
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Let,A={a,b,c} To define a one-one function, For a we have 5 choices and for b we have 4 choices (reason - the function is one - one) And for c we have 3 choices (same reason) Thus , Total no of one one function =5*4*3=60

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You can use permutation here. Then you don't need to care cases that are not one to one.

P(5,3) = $\frac{5!}{2!} = 5 \times 4 \times 3 = 60$

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Answer updated:

There are 5 ways from first element of A to all elements of B, then there are 4 ways for next element of A to B, then there are 3 ways for final element of A to B.

Hence total number of one-to-one functions = 5 × 4 x 3 = 60

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    But the answer is 60 – 2017-01-20
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    How was that obtained? – 2017-01-20
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    @Parul 60 is correct. Did you read it up somewhere or you have an argument? – 2017-01-20
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    Why is 60 correct? – 2017-01-20
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    I read it somewhere.but I didnt understand it. – 2017-01-20
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    Answer updated please review. – 2017-01-20