Mathematics - Functions Question with Solution | TestHub

Let $S=\{1,2,3,4,5,6\}$, then the number of one-one functions, $f:S·P\left(S\right)$, where $P\left(S\right)$ denotes the power set of $S$, such that $f\left(n\right)<f\left(m\right)$ where $n<m$ is $n\left(S\right)=6$

$P\left(S\right)=\left\{\begin{array}{c}\varphi ,\left\{1\right\},\dots \left\{6\right\},\{1,2\},\dots ,\\ \{5,6\},\dots ,\{1,2,3,4,5,6\}\end{array}\right\}$

Case $-1$

$f\left(5\right)=$ any $5$ element subset $A$ of $S$ i.e. $6$ choices

$f\left(4\right)=$ any $4$ element subset $B$ of $A$i.e. $5$  choices

$f\left(3\right)=$ any $3$ element subset $C$ of $B$ i.e. $4$  choices

 $f\left(2\right)=$ any $2$ element subset $D$ of $C$ i.e. $3$  choices

 $f\left(1\right)=$ any $1$ element subset $E$ of $D$ or empty subset i.e. $3$  choices

Total functions $=6\times 5\times 4\times 3\times 3=1080$

Case $-2$

$f\left(6\right)=$ any $5$ element subset $A$ of $S$ i.e. $6$  choices
$f\left(5\right)=$ any $4$ element subset $B$ of $A$ i.e. $5$  choices
$f\left(4\right)=$ any $3$ element subset $C$ of $B$ i.e. $4$  choices
$f\left(3\right)=$ any $2$ element subset $D$ of $C$ i.e. $3$  choices
$f\left(2\right)$ any $1$ element subset $E$ of $D$ i.e. $2$  choices
$f\left(1\right)=$ empty subset i.e. $1$ option

Total functions $=6\times 5\times 4\times 3\times 2\times 1=720$

Case $-3$

$f\left(6\right)=S$

$f\left(5\right)=$ any $4$ element subset $A$ of $S$ i.e. $15$  choices

$f\left(4\right)=$ any $3$ element subset $B$ of $A$ i.e. $4$ choices 
$f\left(3\right)=$any $2$ element subset $C$ of $B$ i.e. $3$ 

 choices

$f\left(2\right)=$ any $1$ element subset $D$ of $C$ i.e. $2$  choices

$f\left(1\right)=$ empty subset i.e. $1$ option

Total functions $=360$

Case $-4$

$f\left(6\right)=S$
$f\left(5\right)=$ any $5$ element subset $A$ of $S$ i.e. $6$  choices
$f\left(4\right)=$ any $3$ element subset $B$ of $A$ i.e. $10$  choices
$f\left(3\right)=$ any $2$ element subset $C$ of $B$ i.e. $3$ options,
$f\left(2\right)=$ any $1$ element subset $D$ of $C$ i.e. $2$ options,
$f\left(1\right)=$ empty subset i.e. $1$ option
Total functions $=360$

Case $-5$

$f\left(6\right)=S$
$f\left(5\right)=$ any $5$ element subset $A$ of $S$ i.e. $6$  choices
$f\left(4\right)=$ any $4$ element subset $B$ of $A$ i.e. $5$  choices
$f\left(3\right)=$any $2$ element subset $C$ of $B$ i.e. $6$  choices
$f\left(2\right)=$ any $1$ element subset $D$ of $C$ i.e. $2$  choices
$f\left(1\right)=$ empty subset i.e. $1$ option

Total functions $=360$

Case $-6$

$f\left(6\right)=S$
$f\left(5\right)=$ any $5$ element subset $A$ of $S$ i.e. $6$  choices
$f\left(4\right)=$ any $4$ element subset $B$ of $A$ i.e. $5$  choices
$f\left(3\right)=$ any $3$ element subset $C$ of $B$ i.e. $4$  choices
$f\left(2\right)=$ any $1$ element subset $D$ of $C$ i.e. $3$  choices
$f\left(1\right)=$ empty subset i.e. $1$ option

Total functions $=360$

$\therefore$ Number of such functions $=3240$