Let S={1,2,3,4,5,6} . Then number of oneone functions f: S → P(S) , where P(S) denote power set of

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1.Relation and Function

hard

Let $S=\{1,2,3,4,5,6\}$. Then the number of oneone functions $f: S \rightarrow P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is $..................$

$3241$

$3242$

$3243$

$3240$

(JEE MAIN-2023)

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 $=3240$

Standard 12

Mathematics

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