5−digit number divisible by 3 formed using 0,1,2,3,4,5 without repetition. Total number of such no.s is ?


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Question Number 50008 by rahul 19 last updated on 13/Dec/18

$5 - d i g i t n u m b e r d i v i s i b l e b y 3 f o r m e d$ $u s i n g 0, 1, 2, 3, 4, 5 w i t h o u t r e p e t i t i o n .$ $T o t a l n u m b e r o f s u c h n o . s i s ?$
Commented by rahul 19 last updated on 13/Dec/18

$5 - d i g i t n u m b e r b e l i k e :$ $\underset{5 w a y s}{-} \underset{5 w a y s}{-} \underset{4 w a y s}{-} \underset{3 w a y s}{-} \underset{}{-}$ $N o w t h e s u m o f s t a r t i n g 4 d i g i t c a n$ $b e o f f o r m 3 k, 3 k + 1, 3 k + 2 .$ $i f 3 k : t h e n l a s t d i g i t c a n b e 0, 3$ $i f 3 k + 1 : t h e n l a s t d i g i t c a n b e 2, 5$ $i f 3 k + 2 : t h e n l a s t d i g i t c a n b e 1, 4 .$ $H e n c e 2 w a y s f o r l a s t d i g i t .$ $∴ N o . o f w a y s = 5 \times 5 \times 4 \times 3 \times 2 = 600$ $M r w s i r, p l t e l l t h e c a s e s w h i c h i$ $o v e r c o u n t . . . i h a v e d i d l i k e Q . 49408 .$ $S i r, i t h i n k t h i s m e t h o d$ $i s a p p l i c a b l e o n l y w h e n r e p e t i t i o n$ $i s a l l o w e d, r i g h t ?$
Commented by mr W last updated on 13/Dec/18

$y o u c a n n o t a p p l y t h i s m e t h o d i f$ $n o r e p e t i t i o n i s a l l o w e d .$
Commented by rahul 19 last updated on 13/Dec/18

$O k a y S i r .$
Commented by mr W last updated on 13/Dec/18

$e . g . i f t h e 4 - d i g i t n u m b e r i s 1230,$ $o r 2301 e t c ., w i t h t h e r e m a i n i n g 4 a n d$ $5 y o u c a n n o t b u i l d a / s u i t a b l e n u m b e r .$ $i f t h e 4 - d i g i t n u m b e r i s 1234, o r 3421$ $e t c ., w i t h t h e r e m a i n i n g 0 a n d 5 y o u$ $c a n o n l y b u i l d o n e s u i t a b l e n u m b e r,$ $n o t t w o a s y o u t o o k . i . e . f r o m t h e$ $300 4 - d i g i t n u m b e r s y o u b u i l d,$ $m a n y o f t h e m a r e i n v a l i d, f r o m e a c h o f$ $t h e o t h e r s y o u c a n o n l y f o r m o n e$ $s u i t a b l e n u m b e r . t h e r e f o r e t h e t o t a l n u m b e r$ $m u s t b e l e s s t h a n 300 .$
	Answered by mr W last updated on 13/Dec/18

	$t o f o r m s u i t a b l e n u m b e r s w e c a n u s e$ $e i t h e r d i g i t s 1 / 2 / 3 / 4 / 5 o r 0 / 1 / 2 / 4 / 5 .$ $\Rightarrow t o t a l n u m b e r o f s u c h n u m b e r s i s$ $5! + 4 \times 4! = 216$
	Commented by mr W last updated on 13/Dec/18

	$a n u m b e r i s d i v i s i b l e b y 3 w h e n t h e$ $s u m o f i t s d i g i t s i s d i v i s i b l e b y 3 .$ $t h e r e f o r e f r o m t h e s i x g i v e n d i g i t s$ $0 t o 5, w e m u s t s e l e c t 5 d i g i t s w h o s e$ $s u m i s d i v i s i b l e b y 3, t h e r e a r e o n l y$ $t w o p o s s i b i l i t i e s :$ $e i t h e r 1 / 2 / 3 / 4 / 5 o r 0 / 1 / 2 / 4 / 5 .$
	Commented by rahul 19 last updated on 13/Dec/18

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