Question

The number of surjections from $A=\left\{1,2,\dots .n\right\},n\ge 2$ onto $B=\left\{a,b\right\}$ is

• A. ${}^{n}P_2$
• B. $2^n-2$
• C. $2^n-1$
• D. None of these

Answer 1

The correct option is B

$2^n-2$

Explanation for the correct option:

Given: $A=\left\{1,2,\dots .n\right\}$ onto $B=\left\{a,b\right\}$

As set $A$ has$n$ elements, set $B$ has $2$ elements.

We know that the number of surjections $=2^n-[$Total number of elements in output set $]$

$\therefore$ No. of surjections is $2^n-2$.

Hence, option(B) is the correct answer.