Let $\mathrm{A}=\{1,2,3,4,5\}$ and $\mathrm{B}=\{1,2,3,4,5,6\}$. Then the number of functions $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ satisfying $\mathrm{f}(1)+\mathrm{f}(2)=\mathrm{f}(4)-1$ is equal to

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Sets Relations and Functions

Let

A = {1, 2, 3, 4, 5}

and

B = {1, 2, 3, 4, 5, 6}

. Then the number of functions

f : A \to B

satisfying

f (1) + f (2) = f (4) - 1

is equal to

Answer: 360

The condition

f (1) + f (2) = f (4) - 1

means

f (4) = f (1) + f (2) + 1

. Since

f (4) \leq 6

, the pair

(f (1), f (2))

must satisfy

f (1) + f (2) \leq 5

with

f (1), f (2) \geq 1

. Ordered pairs of sum

2, 3, 4, 5

number

1 + 2 + 3 + 4 = 10

, and each such pair determines

f (4)

uniquely. The values

f (3)

and

f (5)

are unrestricted, with

6

choices each. Hence the number of functions is

10 \times 6 \times 6 = 360

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Integer answer