Number of onto functions f from {1, 2, 3, …, 20} only {1, 2, 3, …, 20} such that f(k)

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6.Permutation and Combination

hard

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

A

${6^5} \times \left( {15} \right)!$

B

$5! \times 6!$

C

$\left( {15} \right)! \times 6!$

D

${5^6} \times 15$

(JEE MAIN-2019)

Solution

$k = \{ 4,8,12,16,20\} $

$f(k)\,$ can takes the values $\{ 3,6,9,12,15,18\} $

Number of ways ${ = ^6}{C_5}.5!$

$\therefore $ Total number of onto functions

${ = ^6}{C_5}.5!(15!)$

$ = (6!)(15!)$

Standard 11

Mathematics

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