Let $R=\{a, b, c, d, e\}$ and $S=\{1,2,3,4\}$. Total number of onto function $f: R \rightarrow S$ such that $f(a) \neq 1$, is equal to ______.

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Sets Relations and Functions

Let

R = {a, b, c, d, e}

and

S = {1, 2, 3, 4}

. Total number of onto function

f : R \to S

such that

f (a) \neq = 1

, is equal to ______.

Answer: 180

An onto function from the

5

-element set

R

to the

4

-element set

S

sends exactly one pair of elements to a shared value and the remaining three one-to-one. Choosing the shared pair in

(2 5) = 10

ways and assigning the four resulting groups to the four values in

4! = 24

ways,

total onto functions = (2 5) \cdot 4! = 10 \cdot 24 = 240.

By symmetry

f (a)

equals each value of

S

in equally many of these, so

f (a) = 1

\frac{1}{4} \cdot 240 = 60

of them. Hence the number with

f (a) \neq = 1

240 - 60 = 180.

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Integer answer