Question

Assertion :Let $A=\{1,2,3,4,5\}\to B=\{1,2,3,4,5\}$ then number of one-one mapping from $A$ to $B$ such that $f\left(i\right)\ne j$ is equal to $44$ $(1\le i\le 5,1\le j\le 5)$ Reason: The number of dearrengement of $n$ object is given by $n!\{1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+...+\frac{(-1{)}^n}{1!}\}$

• 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 correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Number of mapping which are $1-1$ are the dearrangement of $5$ objects given by

$5![1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}]$

$=5![\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}]$

$=60-20+5-1=44$.
Hence Assertion is followed by Reason.