Let A = a, b, c and B = 1, 2, 3, 4. Then the number of elements in the set C = f : A \to B | 2 \in f(A) and f is not one-one is \underline\hspace2cm .

Functions PYQ — JEE Main 2020 · 5 Sep · Shift 2 · Q28 with Solution

View written solutionFree

Correct answer: 19

We need to count the number of functions $f:A\to B$ where $A=\{a,b,c\},\quad B=\{1,2,3,4\}$ such that:

$2\in f(A)$ , i.e. at least one element of $A$ maps to $2$ .
$f$ is not one-one.

1. Total number of functions from $A$ to $B$

Each of the $3$ elements of $A$ can map to any of the $4$ elements of $B$ . So total functions are $4^3=64.$

2. Count functions with $2\in f(A)$

We want functions where $2$ appears at least once in the image.

Use complement counting:

Total functions: $64$
Functions with no element mapped to $2$ : each element has only $3$ choices $(1,3,4)$

So such functions are $3^3=27.$ Hence functions with $2\in f(A)$ are $64-27=37.$

3. From these, remove the one-one functions

Now count injective functions $f:A\to B$ such that $2\in f(A)$ .

Since $|A|=3$ and $|B|=4$ , the number of one-one functions from $A$ to $B$ is $^4P_3 = 4\cdot 3\cdot 2 = 24.$

Among these, count those containing $2$ in the image. Again use complement:

Total injective functions: $24$
Injective functions avoiding $2$ : then image must lie in $\{1,3,4\}$ , and since domain has $3$ elements, all three must be used.

Number of such injective functions: $3! = 6.$

Therefore injective functions with $2\in f(A)$ are $24-6=18.$

4. Required count: not one-one and $2\in f(A)$

So the required number is $37-18=19.$

5. Final answer

The number of elements in the set $C$ is $\boxed{19}.$

More from Functions

2. Count functions with

2\in f(A)

We want functions where

2

appears at least once in the image.

Use complement counting:

Total functions:

64

Functions with no element mapped to

2

: each element has only

3

choices

(1,3,4)

So such functions are

3^3=27.

Hence functions with

2\in f(A)

are

64-27=37.

3. From these, remove the one-one functions

Now count injective functions

f:A\to B

such that

2\in f(A)

Since

|A|=3

and

|B|=4

, the number of one-one functions from

A

B

^4P_3 = 4\cdot 3\cdot 2 = 24.

Among these, count those containing

2

in the image. Again use complement:

Total injective functions:

24

Injective functions avoiding

2

: then image must lie in

\{1,3,4\}

, and since domain has

3

elements, all three must be used.

Number of such injective functions:

3! = 6.

Therefore injective functions with

2\in f(A)

are

24-6=18.

Functions question

Functions question

1. Total number of functions from $A$ to $B$

2. Count functions with $2\in f(A)$

3. From these, remove the one-one functions

4. Required count: not one-one and $2\in f(A)$

5. Final answer

More from Functions

Functions question

Functions question

1. Total number of functions from $A$ to $B$

2. Count functions with $2\in f(A)$

3. From these, remove the one-one functions

4. Required count: not one-one and $2\in f(A)$

5. Final answer

More from Functions

1. Total number of functions from AAA to BBB

2. Count functions with 2∈f(A)2\in f(A)2∈f(A)

3. From these, remove the one-one functions

4. Required count: not one-one and 2∈f(A)2\in f(A)2∈f(A)

5. Final answer

More from Functions

1. Total number of functions from AAA to BBB

2. Count functions with 2∈f(A)2\in f(A)2∈f(A)

3. From these, remove the one-one functions

4. Required count: not one-one and 2∈f(A)2\in f(A)2∈f(A)

5. Final answer

More from Functions

1. Total number of functions from $A$ to $B$

2. Count functions with $2\in f(A)$

4. Required count: not one-one and $2\in f(A)$

1. Total number of functions from $A$ to $B$

2. Count functions with $2\in f(A)$

4. Required count: not one-one and $2\in f(A)$