Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ such that $1 \in \mathrm{f}(\mathrm{A})$ is equal to :

JEE Main · 2025 · 22 Jan 2025 · Shift 2 · Sets Relations and Functions

Let

A = {1, 2, 3, 4}

and

B = {1, 4, 9, 16}

. Then the number of many-one functions

f : A \to B

such that

1 \in f (A)

is equal to :

(A)127
(B)151✓ Correct
(C)163
(D)139

Answer: (B)

Solution

Total functions from A (4 elements) to B (4 elements):

4^{4} = 256

.

One-one functions:

4! = 24

, so many-one functions total

256 - 24 = 232

.

Count many-one functions with

1 \in / f (A)

: every element of A maps into the remaining 3 elements

{4, 9, 16}

, giving

3^{4} = 81

functions — and since a 4-element domain cannot inject into a 3-element codomain, all of these are automatically many-one.

So many-one functions with

1 \in f (A)

232 - 81 = 151.

JEE Main · 2025 · 22 Jan 2025 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Sets Relations and Functions →

More JEE Main questions on Sets Relations and Functions

The number of relations, defined on the set ${a, b, c, d}$ , which are both reflexive and symmetric, is equal to:
JEE Main 2026View solution →
Let $A = {2, 3, 5, 7, 9}$ . Let R be the relation on A defined by x Ry if and only if $2 x \leq 3 y$ . Let $ℓ$ be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then …
JEE Main 2026View solution →
Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by
$R = {(x, y) : 4 y = 5 x - 3, x, y \in M}$ .
Then the minimum number of elements required to be added in R, in order to make the relation …
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{5 - x}{3 + 2 x}) + \frac{1}{l o g _{e} ( 10 - x )}$ is $(- \infty, α] \cup [β, γ) - {δ}$ , then $6 (α + β + γ + δ)$ is equal to
JEE Main 2026View solution →
The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in Z}$ is
JEE Main 2026View solution →
Let the domain of the function
$f (x) = lo g_{3} lo g_{5} (7 - lo g_{2} (x^{2} - 10 x + 85)) + sin^{- 1} (\frac{3 x - 7}{17 - x})$ be $(α, β]$ . Then $α + β$ is equal to :
JEE Main 2026View solution →