Functions Ques 13
- The number of functions $f$ 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
(2019 Main, 11 Jan II)
(a) $(15)!\times 6!$
(b) $5^6 \times 15$
(c) $5!\times 6!$
(d) $6^5 \times 15$!
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Answer:
Correct Answer: 13.(a)
Solution: (a) According to given information, we have if x = 2
$k \in\{4,8,12,16,20\}$
Then, $f(k) \in\{3,6,9,12,15,18\}$
$ [\because \text { Codomain }(f)=\{1,2,3, \ldots, 20\}] $
Now, we need to assign the value of $f(k)$ for
$k \in{4,8,12,16,20}$ this can be done in ${ }^6 C_5 - 5$
! ways $=6 \cdot 5!=6!$ and remaining 15 elements can be associated by 15! ways.
$\therefore$ Total number of onto functions $= \frac{6!}{6} \cdot S(15,6)$
BETA
