Question

Number of onto(surjective) function from A to B if $n\left( A \right) = 6$ and $n\left( B \right) = 3$ is
A. ${2^6} - 2$
B. ${3^6} - 3$
C. $340$
D. $540$

Answer 1

In order to find the number of onto(surjective) functions between A and B , let us assume the $n\left( A \right) = m$ and $n\left( B \right) = n$ . Since $1 \leqslant n \leqslant m$ , use the direct formula for number of surjective functions as $\Rightarrow \sum\nolimits_{r = 1}^n {{{\left( { - 1} \right)}^{n - r}}\,{}^n{C_r}{r^m}}$ by putting the values of m and n. Simplify the expression to get the required result.

Complete step by step answer:
We are given a relation A to B in which $n\left( A \right) = 6$ and $n\left( B \right) = 3$
$n\left( A \right)$ basically denotes the number of elements in the A set and similarly $n\left( B \right)$ denotes the number of elements in the B set. Let them be $m$ and $n$ respectively.
According to the question, we are supposed to calculate the number of onto or surjective functions that can be created between the elements of set A and B.
So surjective or Onto functions are the functions between A and B in which for every element of B, there is at least one or more than one element matching with A.
To calculate the number of such functions, there is a direct formula for $1 \leqslant n \leqslant m$
No of onto functions=
$\Rightarrow \sum\nolimits_{r = 1}^n {{{\left( { - 1} \right)}^{n - r}}\,{}^n{C_r}{r^m}}$
Putting $m = 6,n = 3$ , we have
$\Rightarrow \sum\nolimits_{r = 1}^3 {{{\left( { - 1} \right)}^{3 - r}}\,{}^3{C_r}{r^6}} \\\ \Rightarrow {\left( { - 1} \right)^2}\,{}^3{C_1}{\left( 1 \right)^6} + {\left( { - 1} \right)^1}\,{}^3{C_2}{\left( 2 \right)^6} + {\left( { - 1} \right)^0}\,{}^3{C_3}{\left( 3 \right)^6} \\\$
Remember the formula of $C\left( {n,r} \right) = {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Simplifying the expression further, we get
$\Rightarrow {\left( 3 \right)^6} - 3 \times {2^6} + 3 = 3\left( {{{\left( 3 \right)}^5} - {2^6} + 1} \right) \\\ \Rightarrow 540 \\\$
So, the correct answer is Option D.

Note: 1. Relation: Let A and B are two sets. Then a relation R from set A to set B is a subset of $A \times B$
Thus, R is a relation from A to B $\Leftrightarrow R \subseteq A \times B$
If R is a relation form a non-void set A to a non-void set B and if $(a,b) \in R$ ,then we write
$a\,R\,b$ which is read as “a is related to b by the relation R “. If $(a,b) \notin R$ , then we write $a\,{R}\,b$ and we say that a is not related to b by the relation R.
2. The formula of number of surjective functions is only value for $1 \leqslant n \leqslant m$ , if $m < n$ , the number of onto functions is 0 as it is not possible to use all elements of $B$ .