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
Case −1
f(5)= any 5 element subset A of S i.e. 6 choices
f(4)= any 4 element subset B of Ai.e. 5 choices
f(3)= any 3 element subset C of B i.e. 4 choices
f(2)= any 2 element subset D of C i.e. 3 choices
f(1)= any 1 element subset E of D or empty subset i.e. 3 choices
Total functions =6×5×4×3×3=1080
Case −2
f(6)= any 5 element subset A of S i.e. 6 choices
f(5)= any 4 element subset B of A i.e. 5 choices
f(4)= any 3 element subset C of B i.e. 4 choices
f(3)= any 2 element subset D of C i.e. 3 choices
f(2) any 1 element subset E of D i.e. 2 choices
f(1)= empty subset i.e. 1 option
Total functions =6×5×4×3×2×1=720
Case −3
f(6)=S
f(5)= any 4 element subset A of S i.e. 15 choices
f(4)= any 3 element subset B of A i.e. 4 choices
f(3)=any 2 element subset C of B i.e. 3
choices
f(2)= any 1 element subset D of C i.e. 2 choices
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 choices
f(4)= any 3 element subset B of A i.e. 10 choices
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 choices
f(4)= any 4 element subset B of A i.e. 5 choices
f(3)=any 2 element subset C of B i.e. 6 choices
f(2)= any 1 element subset D of C i.e. 2 choices
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 choices
f(4)= any 4 element subset B of A i.e. 5 choices
f(3)= any 3 element subset C of B i.e. 4 choices
f(2)= any 1 element subset D of C i.e. 3 choices
f(1)= empty subset i.e. 1 option
Total functions =360
∴ Number of such functions =3240