How many odd numbers with different digits are there from 2019 to 9102?


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Question Number 51227 by mr W last updated on 25/Dec/18

$H o w m a n y o d d n u m b e r s w i t h d i f f e r e n t$ $d i g i t s a r e t h e r e f r o m 2019 t o 9102 ?$
Commented by mr W last updated on 25/Dec/18

$1817 ?$
Commented by Rasheed.Sindhi last updated on 26/Dec/18

$Y o u a r e v e r y r i g h t s i r!$ $S e e m y a n s w e r b e l o w . (T h e$ $i n c o r r e c t a n s w e r h a s b e e n$ $d e l e t e d)$
Commented by mr W last updated on 26/Dec/18

$t h a n k s a l o t s i r f o r c o n f i r m i n g a n d$ $f o r t h e h a r d w o r k!$
Commented by Rasheed.Sindhi last updated on 27/Dec/18

$S i r h o w d i d y o u r e e c h t h e a n s w e r ?$ $P l e a s e s h a r e y o u r a p p r o a c h . Y o u r$ $a p p r o a c h m i g h t b e m o r e e f f i c i e n t!$
	Answered by Rasheed.Sindhi last updated on 26/Dec/18

	$C a s e : T h = 9$ $T h e b i g g e s t n u m b e r i s 9087$ $T h e s m a l l e s t n u m b e r 9013$ $t h = 9 (1 w a y)$ $h = 0 (1 w a y)$ $u = {1, 3, 5, 7, 9} - {0, 9} = {1, 3, 5, 7} (4 w a y s)$ $t = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {0, 9, u} (7 w a y s)$ $∴ T o t a l w a y s (n u m b e r s) w i t h 9 t h o u s a n d$ $1 \times 1 \times 7 \times 4 = 28$ $C a s e : T h = 3, 5, 7$ $t h = {3, 5, 7} (3 w a y s)$ $u = {1, 3, 5, 7, 9} - {t h} (4 w a y s)$ $h = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {t h, u} (8 w a y s)$ $t = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {t h, u, h} (7 w a y s)$ $∴ T o t a l w a y s (n u m b e r s)$ $3 \times 4 \times 8 \times 7 = 672$ $C a s e : T h = 4, 6, 8$ $t h = {4, 6, 8} (3 w a y s)$ $u = {1, 3, 5, 7, 9} - {t h} = {1, 3, 5, 7, 9} (5 w a y s)$ $h = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {t h, u} (8 w a y s)$ $t = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {t h, u, h} (7 w a y s)$ $∴ T o t a l w a y s (n u m b e r s)$ $3 \times 5 \times 8 \times 7 = 840$ $C a s e : T h = 2$ $2013 t o 2987$ $t h = {2} (1 w a y)$ $u = {1, 3, 5, 7, 9} (5 w a y s)$ $h = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, u} (8 w a y s)$ $t = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, u, h} (7 w a y s)$ $∴ T o t a l w a y s (n u m b e r s)$ $1 \times 5 \times 8 \times 7 = 280$ $C a s e : T h = 2 (u p t o 2019 (e x c l u s i v e))$ $t h = {2} (1 w a y), h = {0} (1 w a y)$ $t = {0, 1} - {0} = {1} (1 w a y)$ $u = {1, 3, 5, 7, 9} - {0, 1, 9} = {3, 5, 7} (3 w a y s$ $T o t a l w a y s (n u m b e r s) u p t o 2019 (e x c e p t 2019)$ $1 \times 1 \times 1 \times 3 = 3 (t o b e s u b t r a c t e d)$ $[T h e s e n u m b e r s a r e 2013, 2015 & 2017]$ $T o t a l o f r e q u i r e d n u m b e r s f r o m 2019$ $t o 9102$ $= 28 + 672 + 840 + 280 - 3 = 1817$
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