The probability that a randomly chosen one-one function from the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$ to the set $\{1,2,3,4,5\}$ satisfies $\mathrm{f}(\mathrm{a})+2 \mathrm{f}(\mathrm{b})-\mathrm{f}(\mathrm{c})=\mathrm{f}(\mathrm{d})$ is :

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Probability

The probability that a randomly chosen one-one function from the set

{a, b, c, d}

to the set

{1, 2, 3, 4, 5}

satisfies

f (a) + 2 f (b) - f (c) = f (d)

is :

Answer: (D)

Total one-one functions from a 4-element set to a 5-element set:

n (S) =^{5} C_{4} \times 4! = 120

. For the condition

f (a) + 2 f (b) = f (c) + f (d)

, we find valid assignments:

(5, 1, 3, 4)

(4, 2, 3, 5)

(1, 3, 2, 5)

. Each can be permuted among the four variables in

2

ways (since the equation is symmetric in certain pairs), giving

n (A) = 3 \times 2 = 6

favorable outcomes. Therefore,

P (A) = \frac{6}{120} = \frac{1}{20}

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Single correct option