Question

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

Answer 1

Let A be a set with m elements and B a set of n elements, where m,n are positive integers .

Let    .

If   is a one-to-one function then are distinct element of B so the set B contains at least m elements .

Let's find number of one-to-one functions from A to B .

Let   be any one-to-one function .

Then   have n choises as it can map to any element of B .

Also have n-1 choises as cannot map to the element of B which already mapped by .

have n-2 choises as cannot map to the element of B which already mapped by and   .

and so on .

and al last have n-m+1 choises as cannot map to the element of B which already mapped by ,  ,..., .

Hence total number of one-to-one function from A and B is ,

=  

Note that   is well defined as

Answer :    .

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.If you have any doubt or need more clarification at any step pleae comment.