Question

The number of onto functions from the set {1, 2,.......11} to the set {1,2,......10} is
$\left( a \right)5 \times 11!$
$\left( b \right)10!$
$\left( c \right)\dfrac{{11!}}{2}$
$\left( d \right){\left( {10} \right)^{11}} - 10$

Answer 1

Hint: In this particular question use the concept that if there are two sets having m and n number of elements than if m < n then the number of onto functions are zero, and if m > n then the number of onto functions are ${n^m} - n$ so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Let, A = {1, 2, ........ 11}
And, B = {1, 2,..... 10}
So the number of elements in set A is 11, and the number of elements in set B is 10.
Let, m = 11 and n = 10.
Now we have to find out the number of onto functions from set A to the set B.
Now if there are two sets having m and n number of elements than if m < n then the number of onto functions are zero,
And if m $ \geqslant $ n then the number of onto functions are, ${n^m} - n$.
$ \Rightarrow {\text{onto functions}} = \left\{ {0,m < n} \right.$
                                    =$\left\{ {{n^m} - n,m \geqslant n} \right.$
So as we see that, m $ \geqslant $ n.
So the number of onto functions are ${n^m} - n$
Now, m = 11, and n = 10, so substitute these values in the above equation we have,
\[ \Rightarrow {\text{onto functions}} = {10^{11}} - 10\]
So this is the required answer.

So, the correct answer is “Option d”.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall how to find the onto function of a set to the other set if they have m and n elements in the sets respectively, where m $ \geqslant $n, which is stated above, so simplify substitute the values in the formula we will get the required answer.