Question

If \[f:\left\{ {1,2,3,4,5} \right\} \to \left\{ {1,2,3,4,5} \right\}\;\]those are onto and \[f\left( i \right) \ne i,\]then is equal to
A. \[9\]
B. \[44\]
C. \[16\]
D. None of these

Answer 1

Hint: In the given question, we need to find the number of required functions. For this, we will use the formula of arrangement in mathematics to get the desired result. Before this we will define onto function to understand the question.

Formula used: The following formula used for solving the given question.
The factorial of a number is given by, \[n! = n \times (n - 1) \times .... \times 1\]
Here, \[n\] is a given number.

Complete step by step solution: We know that \[n! = n \times (n - 1) \times .... \times 1\]
Here, onto function is defined as a function \[f\]that maps an element \[x\]to every element \[y\]. That means, for each \[y\], there is an \[x\] such that \[f\left( x \right){\rm{ }} = {\rm{ }}y\].
Thus, the given function contains five elements.
So, we can say that the total number of required functions is the number of rearrangement of these five elements.
Mathematically, we can say that
The total number of desired functions \[ = 5!\left( {\dfrac{1}{{2!}} - \dfrac{1}{{3!}} + \dfrac{1}{{4!}} - \dfrac{1}{{5!}}} \right)\]
This gives the total number of desired functions \[ = 44\]
Thus, the total number of functions are \[44\].

Thus, Option (B) is correct.

Additional Information: A function whose image is the same as its codomain is known as an onto function. Also, an onto function also has an identical range as codomain.

Note: Many students make mistakes in the calculation part as well as arranging the five elements of a given function. This is the only way through which we can solve the example in the simplest way. Also, it is essential to find correct result of \[5!\left( {\dfrac{1}{{2!}} - \dfrac{1}{{3!}} + \dfrac{1}{{4!}} - \dfrac{1}{{5!}}} \right)\].