Question

Assertion :Let $A=\{x_1,x_2,x_3,x_4,x_5\}$ and $B=\{y_1,y_2,y_3\}$. The number of functions from $A$ to $B$ which are not onto is $45$ Reason: The number of onto functions from $A$ to $B$ is equal to the coefficient of $x^5$ in $5!(e^x-1{)}^3$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

No of onto mapping from $A$ to $B$ such that $n\left(A\right)=$ 5, $n\left(A\right)=$ 3
$=$ coefficient of $x^5$ in $5!{(e^x-1)}^3$
$=$ coefficient of $x^5$ in $5![e^{3x}-3e^{2x}+3e^x-1]$
$=5!(\frac{3^5}{5!}-\frac{{3.2}^5}{5!}+3.\frac{1}{5!})$
$=81\times 3-3\times 32+3$
$=3(81-32+1)=150$
$\therefore$ Number of functions which are not onto $=$ Total function -onto function $=3^5-150=193$
$\therefore$ Assertion is false and Reason is true .