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JEE Main 202330 JanuaryMorning Shift
Exam
JEE Main
Year
2023
Date
30 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Functions

JEE Main 2023 · 30 January · Morning Shift · Mathematics · Functions

Let S = 1, 2, 3, 4, 5, 6 . Then the number of oneone functions f: S...

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

Correct answer

3240

Step-by-step solution

Let S=1,2,3,4,5,6S = { 1, 2, 3, 4, 5, 6 }, then the number of one-one functions, f:S⋅P(S)f: S \cdot P ( S ), where P(S)P ( S ) denotes the power set of SS, such that f(n)<f(m)f ( n ) < f ( m ) where n<mn < m is n(S)=6n ( S ) = 6

P\left(S\right) = \left\{ϕ,1,…6,1,2,…,5,6,…,1,2,3,4,5,6\begin{array}{ll}ϕ, { 1 } , \ldots { 6 } , { 1, 2 } , \ldots , \\ { 5, 6 } , \ldots , { 1, 2, 3, 4, 5, 6 }\end{array}\right\}

Case −1- 1

f(5)=f ( 5 ) = any 55 element subset AA of SS i.e. 66 choices

f(4)=f ( 4 ) = any 44 element subset BB of AAi.e. 55 choices

f(3)=f ( 3 ) = any 33 element subset CC of BB i.e. 44 choices

f(2)=f ( 2 ) = any 22 element subset DD of CC i.e. 33 choices

f(1)=f ( 1 ) = any 11 element subset EE of DD or empty subset i.e. 33 choices

Total functions =6×5×4×3×3=1080= 6 \times 5 \times 4 \times 3 \times 3 = 1080

Case −2- 2

f(6)=f ( 6 ) = any 55 element subset AA of SS i.e. 66 choices
f(5)=f ( 5 ) = any 44 element subset BB of AA i.e. 55 choices
f(4)=f\left(4\right) = any 33 element subset CC of BB i.e. 44 choices
f(3)=f ( 3 ) = any 22 element subset DD of CC i.e. 33 choices
f(2)f\left(2\right) any 11 element subset EE of DD i.e. 22 choices
f(1)=f ( 1 ) = empty subset i.e. 11 option

Total functions =6×5×4×3×2×1=720= 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

Case −3- 3

f(6)=Sf ( 6 ) = S

f(5)=f ( 5 ) = any 44 element subset AA of SS i.e. 1515 choices

f(4)=f ( 4 ) = any 33 element subset BB of AA i.e. 44 choices
f(3)=f\left(3\right) =any 22 element subset CC of BB i.e. 33

choices

f(2)=f ( 2 ) = any 11 element subset DD of CC i.e. 22 choices

f(1)=f ( 1 ) = empty subset i.e. 11 option

Total functions =360= 360

Case −4- 4

f(6)=Sf ( 6 ) = S
f(5)=f ( 5 ) = any 55 element subset AA of SS i.e. 66 choices
f(4)=f ( 4 ) = any 33 element subset BB of AA i.e. 1010 choices
f(3)=f\left(3\right) = any 22 element subset CC of BB i.e. 33 options,
f(2)=f\left(2\right) = any 11 element subset DD of CC i.e. 22 options,
f(1)=f ( 1 ) = empty subset i.e. 11 option
Total functions =360= 360

Case −5- 5

f(6)=Sf ( 6 ) = S
f(5)=f ( 5 ) = any 55 element subset AA of SS i.e. 66 choices
f(4)=f ( 4 ) = any 44 element subset BB of AA i.e. 55 choices
f(3)=f\left(3\right) =any 22 element subset CC of BB i.e. 66 choices
f(2)=f ( 2 ) = any 11 element subset DD of CC i.e. 22 choices
f(1)=f ( 1 ) = empty subset i.e. 11 option

Total functions =360= 360

Case −6- 6

f(6)=Sf ( 6 ) = S
f(5)=f ( 5 ) = any 55 element subset AA of SS i.e. 66 choices
f(4)=f ( 4 ) = any 44 element subset BB of AA i.e. 55 choices
f(3)=f\left(3\right) = any 33 element subset CC of BB i.e. 44 choices
f(2)=f ( 2 ) = any 11 element subset DD of CC i.e. 33 choices
f(1)=f ( 1 ) = empty subset i.e. 11 option

Total functions =360= 360

∴∴ Number of such functions =3240= 3240

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