Question

Let A \[=\left\{ {{x}_{1}},{{x}_{2}},.......,{{x}_{7}} \right\}~\]and B \[=\left\{ {{y}_{1}},{{y}_{2}},{{y}_{3}} \right\}~\] be two sets containing seven and three distinct elements respectively. Then the total number of functions f: A→B that are onto. If there exist exactly three elements in A such that f(x) \[={{y}_{2}}\] is equal to:

Answer 1

HINT: - Onto function is a function which has range equal to co-domain which means that every element in the co-domain is mapped by at least one of the elements of the domain.
The formula for selecting r different objects from n different objects is given as follows
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].

Complete step-by-step solution -
As mentioned in the question, for finding the total number of functions which satisfy all the conditions, we follow the following procedure that is
First of all, out of 7, 3 elements from set A are mapped to \[{{y}_{2}}\] . Therefore, for making the functions onto, all the elements from set B have to be mapped and as they all are functions then all elements in the set A also should be mapped.
Hence, the remaining 4 elements of the set A have to be mapped with any of the two elements of set B and for doing this we will use the formula given in the hint as
There are 2 choices for all the 4 elements and we can write that as
\[\begin{align}
  & ={}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}}\times {}^{2}{{C}_{1}} \\
 & =16 \\
\end{align}\]
(By using the fundamental theorem)
Now, the 3 elements which are mapped to \[{{y}_{2}}\] are not known beforehand, so, for choosing 3 elements out of 7 elements in set A, we use the formula mentioned in the hint as follows
\[={}^{7}{{C}_{3}}\]
Now, by fundamental theorem, we get that the total function that are fulfilling all the conditions are
\[={}^{7}{{C}_{3}}\cdot 16\]

NOTE: -
The students can make an error in finding the solution to this question if they don’t know about the fundamental theorem and the definition of onto functions as without the use of these things one cannot get to the correct answer.