Question

One mapping function is selected at random from all the mappings of the set A = {1, 2, 3 … n} into itself. The probability that the mapping selected is one – one is?
$
  {\text{A}}{\text{. }}\dfrac{{{\text{n!}}}}{{{{\text{n}}^{{\text{n - 1}}}}}} \\
  {\text{B}}{\text{. }}\dfrac{{{\text{n!}}}}{{{{\text{n}}^{\text{n}}}}} \\
  {\text{C}}{\text{. }}\dfrac{{{\text{n!}}}}{{{\text{2}}{{\text{n}}^{\text{n}}}}} \\
  {\text{D}}{\text{. None of the above}} \\
$

Answer 1

Hint: In order to find the probability that the mapping is one – one, we first calculate the number of ways mapping can be done in the given set A. Then we calculate the number of ways of mapping one – one. The division of these two terms gives us the answer.

Complete step by step answer:
Given Data,
A = {1, 2, 3 … n}

Now the number of ways the mapping can be done for the Set A into itself is given as, ${{\text{n}}^{\text{n}}}$as each of the element in the set A can be mapped to any value from 1 to n. And there are n elements in total.

Now the number of ways the mapping is one – one is given by,
One – one mapping means each element of the set A is mapped exactly to one element from the list of elements from 1 to n.
Total number of ways of mapping if it is one – one is n!

Now we know the probability of an event is defined by the formula, ${\text{P = }}\dfrac{{{\text{favorable ways}}}}{{{\text{total number of ways}}}}$
Now the probability that the mapping is 1 – 1 is given by,
${\text{P = }}\dfrac{{{\text{n!}}}}{{{{\text{n}}^{\text{n}}}}}$

The probability that the mapping selected is one – one is $\dfrac{{{\text{n!}}}}{{{{\text{n}}^{\text{n}}}}}$
Option B is the correct answer.

Note: In order to solve this type of problem they key is to know how to put out the number of ways the total mapping can be done as well as the total mapping of one – one functions. We have to carefully count the number of ways these can be done. Once we compute this, we put them in the formula of probability to determine the answer.
We have to know the definition of one – one and functions of this type to find the number of ways they can be mapped.