Answer: 3240

Solution:

Formula: Number of One-one functions (3.2)

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

$n(S)=6$

$P(S)={\begin{matrix} \phi,{1}, \ldots{6},{1,2}, \ldots, \\ {5,6}, \ldots,{1,2,3,4,5,6}\end{matrix} }$

• 64 elements

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,

$f^{\prime}(2)=$ 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 - 5

$f(4)=$ any 4 element subset B of A i.e. 5 options,