Let $A=(a, b, c)$ and $B=(1,2,3,4)$. Then the number of elements in the set $C=\{f: A \rightarrow B \mid 2 \in f(A)$ and $f$ is not one-one$\}$ is ____.

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Sets Relations and Functions

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 ____.

Answer: 19

Count functions

f : {a, b, c} \to {1, 2, 3, 4}

with

2

in the range and

f

not one-one. If the range is just

{2}

, all three elements map to

2

: that is

1

function. If the range has exactly two values,

2

together with one of

{1, 3, 4}

, choose the partner in

(1 3) = 3

ways and map three elements onto two values using both,

2^{3} - 2 = 6

ways, giving

3 \times 6 = 18

. A three-element range would force

f

to be one-one, which is excluded. The total is

1 + 18 = 19

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Integer answer