Question

Number of functions from a set $A$ containing $6$ elements to a set $B$ containing $3$ elements such that every element in $B$ has atleast one preimage is

• A. $3^6-{}^3{C}_1-{}^3{C}_2(2^6+2)$
• B. $3^6-{}^3{C}_1-{}^3{C}_2\left(2^6\right)$
• C. $534$
• D. $540$

Answer 1

The correct option is D $540$

We have to find number of functions such that every element of B has a preimage in A
Consider this as a case of $n$ distinct things in $r$ distinct boxes such that each box is filled with atleast one thing.

$=[3^6-3\cdot 2^6+{}^3{C}_2]$
$=[729-192+3]$
$=540$