Question

The total number of injective mappings from a set with m elements to a set with n elements, $m \leqslant n,$ is
$\left( a \right){m^n}$
$\left( b \right){n^m}$
$\left( c \right)\dfrac{{n!}}{{\left( {n - m} \right)!}}$
$\left( d \right)n!$

Answer 1

Hint: In this particular question use the concept that in the injective mapping between two sets if one element of first set is mapped with an element of second set, then the other element of first set cannot mapped with the same element of second set i.e. which is mapped earlier, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
First set with n elements and another set with m elements, where, $n \geqslant m$.
Now we have to find out the total number of injective mappings from a set with m elements to a set with n elements.
Injective mapping
As we all know that injective mapping is also known as one – to – one function i.e. a function that maps distinct elements of its domain to distinct elements of its co-domain, or we can say that every elements of its co-domain is the image of at most one element of its domain.
Now we have to map from a set with m elements to a set with n elements.
So first find out the number of ways to select m elements from n elements as, $n \geqslant m$.
Now as we know that if there are n objects so the number of ways to select r objects out of n is ${}^n{C_r}$.
So the number of ways to select m elements from n elements is ${}^n{C_m}$.
Now we have to arrange these elements so the number of ways to arrange m elements are m!.
So the total number of injective mapping is the product of above two values.
So the total number of injective mapping = ${}^n{C_m}\left( {m!} \right)$
Now as we know that, ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Therefore, total number of injective mapping = $\dfrac{{n!}}{{m!\left( {n - m} \right)!}}\left( {m!} \right) = \dfrac{{n!}}{{\left( {n - m} \right)!}}$
So this is the required answer.
Hence option (c) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is that the number of ways to select r objects out of n is ${}^n{C_r}$ and always recall that the number of ways to arrange n different objects are n!, and always recall that ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$