a challening question: find the number of numbers which are divisible by 9 and consist of distinct digits.


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Question Number 179420 by mr W last updated on 29/Oct/22

$\underset{―}{a c h a l l e n i n g q u e s t i o n :}$ $f i n d t h e n u m b e r o f n u m b e r s w h i c h$ $a r e d i v i s i b l e b y 9 a n d c o n s i s t o f$ $d i s t i n c t d i g i t s .$
Commented by Frix last updated on 01/Nov/22

$I get 59 ‘ ‘ unique {numbers}^{″} which consist$ $of ascending digits without 0$ $9$ $18 27 36 45$ $126 135 189 234 279 369 378 459 468 567$ $1269 1278 1359 1368 1458 1467 2349 2358$ $2367 2457 3456 3789 4689 5679$ $12348 12357 12456 12789 13689 14589$ $14679 15678 23589 23679 24579 24678$ $34569 34578$ $123489 123579 123678 124569 124578$ $134568 156789 234567 246789 345789$ $1236789 1245789 1345689 2345679$ $12345678$ $123456789$ $1 digit 1$ $2 digits 4$ $3 digits 10$ $4 digits 14$ $5 digits 14$ $6 digits 10$ $7 digits 4$ $8 digits 1$ $9 digits 1$ $with all permutations and 0 I get$ $4215386 numbers$
Commented by mr W last updated on 01/Nov/22

$t h a n k s s i r!$ $h o w d i d y o u g e t a l l t h e s e n u m b e r s ?$ $t h e n u m b e r s o f 1 t o 9 d i g i t n u m b e r s$ $a r e c o r r e c t, t h e y c o r r e s p o n d w i t h m y$ $t a b l e .$
Commented by Frix last updated on 01/Nov/22

$I tried to find a system to get them :$ $start with$ $9$ $we can ‘ ‘ {split}^{″} it :$ $18 27 36 45$ $then we can add a 9 :$ $189 279 369 459$ $now sometimes we can ‘ ‘ {shift}^{″} a 1 :$ $i . e . 369 \Rightarrow 378$ $and so on$ $if we always stay with ascending digits {it}^{'} s$ $a fast way$
Commented by mr W last updated on 01/Nov/22

$t h a n k s a l o t f o r e x p l a i n i n g!$
	Answered by Rasheed.Sindhi last updated on 29/Oct/22

	$\begin{matrix} A T ry . . . \end{matrix}$ $C - 9 : Sum of digits = 9$ $A l l p e r m u t a t i o n s o f$ $p a r t i o n s o f 9 c o n s i s t o f d i s t i n c t d i g i t s$ $w h i c h a r e n o t s t a r t i n g w i t h 0 f r o m l e f t .$ $a > b > c > d \Rightarrow 3 ⩽ a ⩽ 9 \land 6 ⩽ d ⩽ 0$ $0 : 1 + 8, 2 + 7, 3 + 6, 4 + 5, . . .$ $1 :$ $C - 18 : Sum of digits = 18$ $C - 27 : Sum of digits = 27$ $C - 36 : Sum of digits = 36$ $C - 45 : Sum of digits = 45$ $. . .$
	Commented by mr W last updated on 30/Oct/22

	$i t s e e m s n o o n e t a k e s t h e c h a l l e n g e$ $e x c e p t y o u . t h a n k s f o r t r y i n g s i r!$ $p l e a s e c o n t i n u e s i r!$ $i^{'} l l a l s e g i v e a n a t t e m p t .$
	Commented by Rasheed.Sindhi last updated on 30/Oct/22

	$S i r a c t u a l l y a t t h e m o m e n t I {d o n}^{'} t$ $k n o w f o r m u l a s o f n u m b e r o f p a r t i t i o n s /$ $\sim w i t h s o m e r e s t r i c t i o n s! E v e n I$ $a m n o t s u r e o f m y a p p r o a c h t o b e$ $c o r r e c t!$ $I t h i n k m y a p p r o a c h i s c l e a r a t l e a s t .$ $(i) D i g i t s u m o f s u c h n u m b e r s i s$ $b e t w e e n 9 & 45 i n c l u s i v e l y a n d m u s t$ $b e d i v i s i b l e b y 9 i - e i t m a y b e e q u a l$ $t o 9, 18, 27, 36 o r 45 . S o s u c h n u m b e r s$ $m u s t b e p a r t i t i o n s o f t h e s e n u m b e r s$ $w i t h t w o m o r e r e s t r i c t i o n s : b e i n g$ $d i s t i n c t a n d n o t b e g i n n i n g w i t h 0$ $(f r o m l e f t) .$ $C o u l d y o u p l e a s e s a y s o m e t h i n g$ $a b o u t t h i s a p p r o a c h . (O f c o u r s e t h e r e$ $m a y b e b e t t e r a p p r o a c h e s) b u t i s$ $t h i s a w r o n g a p p r o a c h ?$
	Commented by mr W last updated on 30/Oct/22

	$y o u r i d e a i s a b s o l u t e l y c o r r e c t . i t h i n k$ $t h e r e a r e n o o t h e r s h o r t c u t m e t h o d s$ $t h a n t h i s . s o t h e t a s k i s t o f i n d t h e$ $n u m b e r o f w a y s t o s e l e c t r d i g i t s f r o m$ $[1, 9] s u c h t h a t t h e i r s u m i s a m u l t i p l e$ $o f 9, i . e . 9, 18, 27, 36, 45 .$ $h e r e r = 2, 3, . . ., 9 .$
	Commented by Rasheed.Sindhi last updated on 30/Oct/22

	$G rateful sir!$
	Commented by Rasheed.Sindhi last updated on 30/Oct/22

	$S o r r y s i r, I^{'} v e o n l y r o u g h i d e a .$ $C o m p l e t e s o l u t i o n i s b e y o n d m y$ $v e r y l i m i t e d c a p a c i t y!$
	Commented by mr W last updated on 31/Oct/22

	$t h a n k s f o r t i l l n o w! p l e a s e c h e c k m y$ $a n s w e r s i r!$
	Answered by mr W last updated on 31/Oct/22

	$a n a t t e m p t$ $s i n c e s u c h a n u m b e r s h o u l d c o n s i s t$ $o f d i s t i n c t d i g i t s, i t m u s t h a v e a t$ $l e a s t t w o d i g i t s a n d a t m o s t 10 d i g i t s .$ ${l e t}^{'} s c o n s i d e r a t f i r s t o n l y n u m b e r s$ $w i t h o u t t h e d i g i t ‘ ‘ 0^{″}, s i n c e i f w e h a v e$ $a r - d i g i t - n u m b e r w h i c h i s d i v i s i b l e$ $b y 9, b u t w i t h o u t ‘ ‘ 0^{″}, t h e n w e h a v e$ $a u t o m a t i c a l l y r$ $(r + 1) - d i g i t - n u m b e r s w h i c h a r e$ $a l s o d i v i s i b l e b y 9 . a s e x a m p l e :$ $s a y d_{1} d_{2} . . . d_{r} i s d i v i s i b l e b y 9, w i t h d_{i} \neq 0,$ $t h e n d_{1} 0 d_{2} . . . d_{r}, d_{1} d_{2} 0 . . . d_{r}, . . ., d_{1} d_{2} . . . d_{r} 0$ $a r e a l s o d i v i s i b l e b y 9 .$ $w e k n o w a n u m b e r i s d i v i s i b l e b y 9,$ $i f t h e s u m o f i t s d i g i t s i s d i v i s i b l e b y$ $9 . t h e s u m o f 9 d i f f e r e n t d i g i t s c a n$ $a t m o s t b e$ $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 .$ $t h e r e f o r e t h e s u m o f t h e d i g i t s f r o m$ $a n u m b e r w i t h d i s t i n c t d i g i t s a n d$ $d i v i s i b l e b y 9 m u s t b e$ $9 o r 18 o r 27 o r 36 o r 45 .$ $n o w w e s h o u l d f i n d t h e n u m b e r o f$ $w a y s t o s e l e c t r d i g i t s f r o m [1, 9]$ $s u c h t h a t t h e i r s u m i s 9 o r 18 o r 27 o r$ $36 o r 45 . h e r e r = 1, 2, . . ., 9 .$ $s a y t h e r d i s t i n c t d i g i t s a r e$ $d_{1}, d_{2}, . . ., d_{r} w i t h$ $d_{1} < d_{2} < . . . < d_{r} a n d 1 ⩽ d_{i} ⩽ 9$ $w i t h t h e m w e c a n f o r m r!$ $r - d i g i t - n u m b e r s .$ $d_{1} + d_{2} + . . . + d_{r} = 9 o r 18 o r 27 o r 36 o r 45 .$ $t h e n u m b e r o f i n t e g e r s o l u t i o n s o f$ $t h i s e q u a t i o n r e p r e s e n t s t h e n u m b e r$ $o f w a y s t o s e l e c t d_{1}, d_{2}, . . ., d_{r} f r o m [1, 9] .$
	Commented by mr W last updated on 31/Oct/22

	$n u m b e r o f w a y s t o s e l e c t r d i g i t s f r o m$ $[1, 9] s u c h t h a t t h e i r s u m i s e q u a l t o . . .$
	Commented by mr W last updated on 01/Nov/22

	Commented by mr W last updated on 01/Nov/22

	$2 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : 4 \times 2! = 8$ $w i t h ‘ ‘ 0^{″} : 1 \times 1 = 1$ $3 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : (3 + 7) \times 3! = 60$ $w i t h ‘ ‘ 0^{″} : 8 \times 2 = 16$ $4 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : (11 + 3) \times 4! = 336$ $w i t h ‘ ‘ 0^{″} : 60 \times 3 = 180$ $5 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : (3 + 11) \times 5! = 1 680$ $w i t h ‘ ‘ 0^{″} : 336 \times 4 = 1 344$ $6 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : (7 + 3) \times 6! = 7 200$ $w i t h ‘ ‘ 0^{″} : 1680 \times 5 = 8 400$ $7 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : 4 \times 7! = 20 160$ $w i t h ‘ ‘ 0^{″} : 7200 \times 6 = 43 200$ $8 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : 1 \times 8! = 40 320$ $w i t h ‘ ‘ 0^{″} : 20160 \times 7 = 141 120$ $9 - d i g i t n u m b e r s :$ $w i t h o u t ‘ ‘ 0^{″} : 1 \times 9! = 362 880$ $w i t h ‘ ‘ 0^{″} : 40320 \times 8 = 322 560$ $10 - d i g i t n u m b e r s :$ $w i t h ‘ ‘ 0^{″} : 362880 \times 9 = 3 265 920$ $t o t a l l y : 4 215 385 n u m b e r s$
	Commented by mr W last updated on 31/Oct/22

	$t o c h e c k t h e t a b l e a b o v e w e c a n$ $c a l c u l a t e t h e t o t a l n u m b e r (Σ) o f$ $w a y s t o s e l e c t 1, 2, . . ., 9 t o g e t t h e s u m k .$ $i t i s t h e c o e f f i c i e n t o f x^{k} i n t h e$ $e x p a n s i o n o f$ $(1 + x) (1 + x^{2}) (1 + x^{3}) . . . (1 + x^{9}) .$ $h e r e w e o n l y n e e d t o l o o k a t t h e v a l u e s$ $f o r k = 9, 18, 27, 36, 45 .$
	Commented by mr W last updated on 31/Oct/22

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