Question

Let A be a set of 4 elements. From the set of all functions A to A, the probability that it is an into function is?
(a) $\dfrac{3}{32}$
(b) 0
(c) $\dfrac{29}{32}$
(d) 1

Answer 1

We start solving the problem by finding the total number of functions that can be formed from A to A using the fact that the total number of functions that can be formed from a set C to C where C is having ‘n’ elements is ${{n}^{n}}$ functions. We then find the total number of functions that can be formed from A to A using the fact that the total number of into functions that can be formed from a set C to C where C is having ‘n’ elements is ${{n}^{n}}-n!$ functions. We then substitute the obtained values in the formula probability = $\dfrac{\text{Total number of into functions from A to A}}{\text{Total number of functions from A to A}}$ to get the required answer.

Complete step by step answer:
According to the problem, we are given that A is a set consisting of 4 elements. We need to find the probability that the set of functions from A to A is an into function.
Let us find the total number of functions from sets A to A can be formed. We know that the total number of functions that can be formed from a set C to C where C is having ‘n’ elements is ${{n}^{n}}$ functions.
So, the total number of functions that can be formed from A to A is ${{4}^{4}}=256$ functions ---(1).
We know that the total number of into functions that can be formed from a set C to C where C is having ‘n’ elements is ${{n}^{n}}-n!$ functions, where $n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times ......\times 3\times 2\times 1$.
So, the number of into functions that can be formed from A to A is ${{4}^{4}}-4!=256-\left( 4\times 3\times 2\times 1 \right)=256-24=232$ functions ---(2).
Now, let us find the probability that the functions from A to A is an into function.
We know that probability = $\dfrac{\text{Total number of into functions from A to A}}{\text{Total number of functions from A to A}}$.
$\Rightarrow$ Probability = $\dfrac{232}{256}=\dfrac{29}{32}$.
We have found the required probability as $\dfrac{29}{32}$.

So, the correct answer is “Option c”.

Note: We should not confuse into function with the onto function while solving this problem. We can also find the total number of into functions by subtracting the total number of onto functions from the total number of functions. We should know that the value of probability is between 0 and 1. Similarly, we can expect problems to find the probability that the functions from A to A are many to one function.