Question

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

Answer 1

The correct answer is 3240.
Let S={1,2,3,4,5,6}, then the number of one-one functions, f:S⋅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)={ϕ,{1},…{6},{1,2},…,{5,6},…,{1,2,3,4,5,6}​}
−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′(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 −5
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 2 element subset C of B i.e. 6 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 suhset 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
∴ Number of surch funstions =3240