How many 5 digit numbers with different digits are multiple of 9?


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Question Number 178395 by mr W last updated on 16/Oct/22

$H o w m a n y 5 d i g i t n u m b e r s w i t h$ $d i f f e r e n t d i g i t s a r e m u l t i p l e o f 9 ?$
	Answered by cortano1 last updated on 16/Oct/22

	$(1) 1 + 5 + 6 + 7 + 8 \Rightarrow 5!$ $(2) 1 + 2 + 3 + 4 + 8 \Rightarrow 5!$ $(3) 1 + 2 + 4 + 5 + 6 \Rightarrow 5!$ $(4) 1 + 2 + 3 + 5 + 7 \Rightarrow 5!$ $(5) 1 + 3 + 6 + 8 + 9 \Rightarrow 5!$ $(6) 1 + 4 + 5 + 8 + 9 \Rightarrow 5!$ $(7) 1 + 2 + 7 + 8 + 9 \Rightarrow 5!$ $(8) 2 + 3 + 5 + 8 + 9 \Rightarrow 5!$ $\Rightarrow 8 \times 5! = 8 \times 120 = 960$
	Commented by mr W last updated on 16/Oct/22

	$w h a t a b o u t n u m b e r s l i k e 13068, 47981$ $e t c . ?$
	Commented by Tawa11 last updated on 16/Oct/22

	$Great sir$
	Commented by mr W last updated on 16/Oct/22

	$i m e a n t i n h i s s o l u t i o n n u m b e r s l i k e$ $13068, 47981 e t c . a r e n o t i n c l u d e d .$
	Commented by mr W last updated on 16/Oct/22

	$s o a n s w e r 960 i s n o t c o r r e c t .$
	Commented by cortano1 last updated on 16/Oct/22

	$do you meant the numbers is$ $formed from 0, 1, . . ., 9 ?$
	Commented by mr W last updated on 16/Oct/22

	$y e s . a l l d i g i t s a r e a l l o w e d .$
	Commented by cortano1 last updated on 16/Oct/22

	$ok sir$
	Commented by mr W last updated on 16/Oct/22

	$t y p o . {i t}^{'} s j u s t a n e x a m p l e .$
	Commented by cortano1 last updated on 16/Oct/22

	$what your answer sir ?$
	Commented by mr W last updated on 16/Oct/22

	$i g o t 3024 .$
	Answered by mr W last updated on 16/Oct/22

	$0 + 1 + 2 + 3 + 4 = 10$ $9 + 8 + 7 + 6 + 5 = 35$ $t h a t m e a n s t h e s u m o f 5 d i f f e r e n t$ $d i g i t s l i e s b e t w e e n 10 a n d 35 .$ $s u c h t h a t a 5 d i g i t n u m b e r i s$ $m u l t i p l e o f 9, t h e s u m o f i t s d i g i t s$ $m u s t b e 27 o r 18 .$ $c a s e 1 : s u m o f d i g i t s = 27$ $9 + 8 + 7 + 3 + 0 \Rightarrow \frac{4}{5} \times 5! n u m b e r s$ $9 + 8 + 6 + 4 + 0$ $9 + 7 + 6 + 5 + 0$ $9 + 8 + 7 + 2 + 1 \Rightarrow 5! n u m b e r s$ $9 + 8 + 6 + 3 + 1$ $9 + 8 + 5 + 4 + 1$ $9 + 8 + 5 + 3 + 2$ $9 + 7 + 6 + 4 + 1$ $9 + 7 + 6 + 3 + 2$ $9 + 7 + 5 + 4 + 2$ $9 + 6 + 5 + 4 + 3$ $8 + 7 + 6 + 5 + 1$ $8 + 7 + 6 + 4 + 2$ $8 + 7 + 5 + 4 + 3$ $\Rightarrow 3 \times \frac{4}{5} \times 5! + 11 \times 5!$ $c a s e 2 : s u m o f d i g i t s = 18$ $9 + 6 + 2 + 1 + 0 \Rightarrow \frac{4}{5} \times 5! n u m b e r s$ $9 + 5 + 3 + 1 + 0$ $9 + 4 + 3 + 2 + 0$ $8 + 7 + 2 + 1 + 0$ $8 + 6 + 3 + 1 + 0$ $8 + 5 + 4 + 1 + 0$ $8 + 5 + 3 + 2 + 0$ $7 + 6 + 4 + 1 + 0$ $7 + 6 + 3 + 2 + 0$ $7 + 5 + 4 + 2 + 0$ $6 + 5 + 4 + 3 + 0$ $8 + 4 + 3 + 2 + 1 \Rightarrow 5! n u m b e r s$ $7 + 5 + 3 + 2 + 1$ $6 + 5 + 4 + 2 + 1$ $\Rightarrow 11 \times \frac{4}{5} \times 5! + 3 \times 5!$ $t o t a l l y :$ $\Rightarrow (11 + 3) (1 + \frac{4}{5}) \times 5! = 14 \times 9 \times 4! = 3024$
	Commented by SLVR last updated on 25/Oct/22

	$i t i s m o r e c l e a r . . . a w e s o m . s u p e r$
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