The number of functions

Question:

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

  1. $(15) ! \times 6 !$

  2. $5^{6} \times 15$

  3. $5 ! \times 6 !$

  4. $6^{5} \times(15) !$


Correct Option: 1

Solution:

$f(\mathrm{k})=3 \mathrm{~m}(3,6,9,12,15,18)$

for $k=4,8,12,16,20$                     $6.5 .4 .3 .2$ ways

For rest numbers 15 ! ways

Total ways $=6 !(15 !)$

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