The number of functions from $\{1,2,3, \ldots, 20\}$ onto $\{1,2,3, \ldots, 20\}$ such that $f(k)$ is a multiple of 3, whenever k is a multiple of 4, is:

JEE Main · 2019 · 11 Jan 2019 · Shift 2 · Permutations and Combinations

The number of functions from

{1, 2, 3, \dots, 20}

onto

{1, 2, 3, \dots, 20}

such that

f (k)

is a multiple of 3, whenever k is a multiple of 4, is:

Answer: (C)

The map is a bijection of

{1, \dots, 20}

onto itself. The multiples of

4

are

{4, 8, 12, 16, 20}

— five elements — and they must be sent into the multiples of

3

{3, 6, 9, 12, 15, 18}

— six elements. Choosing distinct images for the five from the six available can be done in

6 \times 5 \times 4 \times 3 \times 2 = 6!

ways, and the remaining

15

elements are then matched bijectively with the

15

leftover targets in

15!

ways. The total count is

15! \times 6!

JEE Main · 2019 · 11 Jan 2019 · Shift 2 · Single correct option