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माना $\mathrm{S}=\{1,2,3,4,5,6\}$ है तो ऐसे ऐकेकी फलनों $\mathrm{f}: \mathrm{S} \rightarrow \mathrm{P}(\mathrm{S})$, जहाँ $\mathrm{P}(\mathrm{S})$ समुच्चय $\mathrm{S}$ का घात समुच्चय $\mathrm{f}(\mathrm{n}) \subset \mathrm{f}(\mathrm{m})$ है जब भी $\mathrm{n}<\mathrm{m}$ है, की संख्या है_______.
$3241$
$3242$
$3243$
$3240$
Solution
Let $S=\{1,2,3,4,5,6\}$, then the number of one-one functions, $f : S \cdot P ( S )$, where $P ( S )$ denotes the power set of $S$, such that $f ( n ) < f ( m )$ where $n < m$ is
$n(S)=6$
$P(S)=\left\{\begin{array}{c}\phi,\{1\}, \ldots\{6\},\{1,2\}, \ldots, \\\{5,6\}, \ldots,\{1,2,3,4,5,6\}\end{array}\right\}$
case $-1$
$f(6)=S$ i.e. $1$ option,
$f(5)=$ any $5$ element subset $A$ of $S$ i.e. $6$ options,
$f(4)=$ any $4$ element subset $B$ of $A$ i.e. $5$ options,
$f (3)=$ any $3$ element subset $C$ of $B$ i.e.$4$ options,
$f (2)=$ any $2$ element subset $D$ of $C$ i.e. $3$ options,
$f (1)=$ any $1$ element subset $E$ of $D$ or empty subset i.e. $3$
options,Total functions $=1080$
Case $-2$
$f (6)=$ any $5$ element subset $A$ of $S$ i.e. $6$ options,
$f(5)=$ any $4$ element subset $B$ of $A$ i.e. $5$ options,
$f^{\prime}(4)=$ any $3$ element subset $C$ of $B$ i.e. $4$ options,
$f (3)=$ any $2$ element subset $D$ of $C$ i.e. $3$ options,
$f' (2) =$ any $1$ element subset $E$ of $D$ i.e. $2$ options,
$f(1)=$ empty subset i.e.$1$ option Total functions $=720$
Case $-3$
$f(6)=S$
$f(5)=$ any $4$ element subset $A$ of' $S$ i.e. $15$ options,
$f(4)=$ any $3$ element subset $B$ of $A$ i.e. $4$ options,
$f(3)=$ any $2$ element subset $C$ of B i.e. $3$ options,
$f(2)=$ any $1$ element subset $D$ of $C$ i.e. $2$ options,
$f (1)=$ empty subset i.e. $1$ option
Total functions $=360$
Case $-4$
$f(6)=S$
$f(5)=$ any $5$ element subset $A$ of $S$ i.e. $6$ options,
$f(4)=$ any $3$ element subset $B$ of $A$ i.e. $10$ options,
$f(3)=$ any $2$ element subset $C$ of $B$ i.e. $3$ options,
$f(2)=$ any $1$ element subset $D$ of $C$ i.e. $2$ options,
$f(1)=$ empty subset i.e. $1$ option
Total functions $=360$
Case $-6$
$f (6)= S$
$f (5)=$ any $5$ element subset $A$ of $S$ i.e. $6$ options,
$f (4)=$ any $4$ element subset $B$ of $A$ i.e. $5$ options,
$f (3)=$ any $3$ element subset $C$ of $B$ i.e. $4$ options,
$f (2)=$ any $1$ element subset $D$ of $C$ i.e. $3$ options,
$f (1)=$ empty subset i.e. 1 option
Total functions $=360$
$\therefore$ Number of such functions $=3240$
