Question

$X$ is a set of $3$ digit numbers divisible by $6$ and $Y$ is a set of $3$ digit numbers divisible by $4$, using the digits $0,1,2,3$ without repetition. The number of onto functions from $X$ to $Y$ is

• A. $240$
• B. $244$
• C. $784$
• D. $1024$

Answer 1

The correct option is A $240$

$X=\{102,120,132,210,312\}$
$Y=\{120,132,312,320\}$

Total number of onto functions
$=\sum _{k=0}^n(-1{)}^k{}^n{C}_k(n-k{)}^m$
where $n$ is the number of elements in domain,
$m$ is the number of elements in co-domain.

Here, $m=5,n=4$
Therefore, total number of onto functions $=4^5-{(}^4{C}_1\left)\right(4-1{)}^5+{(}^4{C}_2\left)\right(4-2{)}^5-{(}^4{C}_3\left)\right(4-3{)}^5$
$=240$