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Question

The total number of injective mappings from a set with m elements to a set with n elements, mn, is


A

mn

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B

nm

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C

n!(n-m)!

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D

n!

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Solution

The correct option is C

n!(n-m)!


Explanation for the correct option:

Option (C):

Given data:

  • First set with n elements and another set with m elements, where, nm.
  • 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 element 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, nm.
  • Now as we know that if there are n objects so the number of ways to select r objects out of n is nCr.
    So the number of ways to select m elements from n elements is nCm.
  • Now we have to arrange these elements so the number of ways to arrange m elements is m!.
    So the total number of injective mapping is the product of the above two values.
  • So the total number of injective mapping =nCm(m!)
    Now as we know that,nCr=n!r!(nr)!
    Therefore, the total number of injective mapping =n!m!(nm)!(m!)=n!(nm)!
    So this is the required answer.
    Hence option (c) is the correct answer

Hence option (c) is the correct answer.


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