Question

How many one-to-one functions are there from a set m elements to one with n elements

Answer 1

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Answer 2

Concept:

Function describes a relation between inputs and outputs. In this each input is related to each and unique outputs.

Given:

Set $m$ elements to one with $n$ elements.

Find:

Number of one-to-one functions are there in the set of $m$ elements to one with $n$ elements.

Solution:

According to the problem,

For $m$ elements to one with $n$ elements,

When, $m > n$ aren't any.

When $m\leq n$, there are $n$ options for where to send the first element, and $n-1$ options for the second element, $n-2$ for the third element and so on.

Thus, the total number of $1:1$ functions for $m$ set to an $n$ set is,

$n(n-1)(n-2).......(n-m+1)=\frac{n!}{(n-m)!}$

Hence, the number of one-to-one fnctions are $n(n-1)(n-2).......(n-m+1)=\frac{n!}{(n-m)!}$