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

Hint: 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 $ .