Question

If $P=\left\{1,2,3,4,5\right\}$ and $Q=\left\{a,b,c\right\}$ , then the number of onto functions from $P$ to $Q$ is

• A. $150$
• B. $144$
• C. $147$
• D. $154$

Answer 1

Answer: (A)

Total number of functions $=3^5$
(since each of $1,2,3,4$ or $5$ can correspond to any of $a,b$ or $c$ )
The number of functions that corresponds to only one element of $B$ is ${}^3{C}_1\times 1^5$ and the number of functions that correspond to atmost two elements of $B$ is ${}^3{C}_2\times 2^5.$
Total number of onto functions $=3^5{-}^3{C}_2\times 2^5{+}^3{C}_1\times 1^5=243-96+3=150$
$\left({}^3{C}_1\times 1^{5}isrepeatedtwicein{\left(\right)}^3{C}_2\times 2^5\right)$