help How many onetoone functions are there from a set of car

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How many one-to-one functions are there from a set of cardinality 8 to sets with the following cardinalities? 6, 8, 10, 12.

Solution

Suppose XX has NN elements and YY has MM elements.

For the one-to-one function, each element in X is mapped to a unique element in Y.
Therefore, there are M ways to map the first element in X, and M1 ways to map the second one, etc.
There should be totally M!/(MN)! ways of one-to-one mapping when MN.
When M<N, you cannot get any one-to-one mapping.

Given:

Number of elements in X = 8

a)
   If number of elements in Y = 6

   Zero, because there are not enough elements in Y to map each element of X uniquely

B)
   If number of elements in Y = 8

   Nuber of one to one = 8!/0! = 8!

c)

   If number of elements in Y = 10

   Nuber of one to one = 8!/2! = 3x4x5x6x7x8

d)

   If number of elements in Y = 12

   Nuber of one to one = 8!/4! = 5x6x7x8