How many odd numbers, greater than 60000, can be made from the digits 5,6,7,8,9,0, if no number contains any digit more than ones?


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Question Number 56688 by pieroo last updated on 21/Mar/19

$How many odd numbers, greater than$ $60000, can be made from the digits$ $5, 6, 7, 8, 9, 0, if no number contains any$ $digit more than ones ?$
	Answered by mr W last updated on 22/Mar/19

	$6 X X X Y, 8 X X X Y : 2 \times 3 \times P_{3}^{4}$ $7 X X X Y, 9 X X X Y : 2 \times 2 \times P_{3}^{4}$ $Z X X X X Y : 3 \times 4 \times 4!$ $\Rightarrow (2 \times 3 + 2 \times 2) P_{3}^{4} + 3 \times 4 \times 4! = 528$
	Commented by malwaan last updated on 22/Mar/19

	$I d o n t u n d e r s t a n t$ $p l e a s e m o r e e x p l a i n a s i o n$
	Commented by rahul 19 last updated on 22/Mar/19

	$6 \underset{\underset{4 w a y s}{↓}}{-} \underset{\underset{3 w a y s}{↓}}{-} \underset{\underset{2 w a y s}{↓}}{-} \underset{\underset{^{3} C_{1}}{↓}}{-}$ $s i m i l a r l y f o r d i g i t s s t a r t i n g w i t h 8 .$ $7 \underset{\underset{4 w a y s}{↓}}{-} \underset{\underset{3 w a y s}{↓}}{-} \underset{\underset{2 w a y s}{↓}}{-} \underset{\underset{^{2} C_{1}^{}}{↓}}{-}$ $S i m i l a r l y f o r d i g i t s s t a r t i n g w i t h 9 .$ $\Rightarrow 2 (4 \times 3 \times 2 \times 3) + 2 (4 \times 3 \times 2 \times 2) = 240 .$ $(D e a r p i e r o o,$ $i n t h i s {s o l}^{n} i^{'} v e c o u n t e d 5 - d i g i t n o . s$ $o n l y! S o t h i s {s o l}^{n} i s i n c o m p l e t e .)$
	Commented by mr W last updated on 22/Mar/19

	$t h i s i s c o r r e c t f o r 5 - d i g i t n u m b e r s .$ $b u t y o u c a n a l s o m a k e 6 - d i g i t n u m b e r s .$ $t h e y a r e a l l g r e a t e r t h a n 60000 .$
	Commented by mr W last updated on 22/Mar/19

	$w i t h t h e 6 d i g i t s w e c a n m a k e 5 - d i g i t$ $n u m b e r s (l i k e 78065) a n d 6 - d i g i t$ $n u m b e r s (l i k e 567809) .$ $5 - d i g i t n u m b e r s g r e a t e r t h a n 6000 :$ $6 X X X Y : Y c a n b e 5, 7, 9 \Rightarrow 3 w a y s, X X X f r o m t h e r e m a i n i n g 4 d i g i t s \Rightarrow P_{3}^{4} w a y s, t o t a l l y 3 \times P_{3}^{4} w a y s .$ $8 X X X Y : a s 6 X X X Y, t o t a l l y 3 \times P_{3}^{4} w a y s .$ $7 X X X Y : Y c a n b e 5, 9 \Rightarrow 2 w a y s, X X X f r o m t h e r e m a i n i n g 4 d i g i t s \Rightarrow P_{3}^{4} w a y s, t o t a l l y 2 \times P_{3}^{4} w a y s .$ $8 X X X Y : a s 7 X X X Y, t o t a l l y 2 \times P_{3}^{4} w a y s .$ $\Rightarrow 5 - d i g i t n u m b e r s : 2 \times 3 \times P_{3}^{4} + 2 \times 2 \times P_{3}^{4} = 240$ $6 - d i g i t n u m b e r s :$ $Z X X X X Y :$ $Y c a n b e 5, 7, 9 \Rightarrow 3 w a y s$ $Z c a n b e o n e o f t h e 4 r e m a i n i n g (0 n o t a l l o w e d) \Rightarrow 4 w a y s$ $X X X X f r o m t h e 4 r e m a i n i n g d i g i t s \Rightarrow 4! w a y s$ $\Rightarrow 6 - d i g i t n u m b e r s : 3 \times 4 \times 4! = 288$ $a l l p o s s i b l e n u m b e r s : 240 + 288 = 528$
	Commented by rahul 19 last updated on 22/Mar/19

	$T h a n k Y o u S i r! :)$
	Commented by malwaan last updated on 28/Mar/19

	$t h a n k y o u s o m u c h s i r$
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