Let S={1, 2, 3, 4, 5} , we want to make a group H of numbers have the following properties: 1. Each number has different digits and taken from S 2. Each number is greater than 20 000 3. None of them is multiple of five How many items of H?


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Question Number 178021 by Acem last updated on 12/Oct/22

$L e t S = {1, 2, 3, 4, 5}, w e w a n t t o m a k e a$ $g r o u p H o f n u m b e r s h a v e t h e f o l l o w i n g$ $p r o p e r t i e s :$ $1 . E a c h n u m b e r h a s d i f f e r e n t d i g i t s a n d$ $t a k e n f r o m S$ $2 . E a c h n u m b e r i s g r e a t e r t h a n 20 000$ $3 . N o n e o f t h e m i s m u l t i p l e o f f i v e$ $H o w m a n y i t e m s o f H ?$
	Answered by mr W last updated on 12/Oct/22

	$4 \times 4 \times 3 \times 2 \times 1 - 3 \times 3 \times 2 \times 1 = 78$
	Commented by Tawa11 last updated on 12/Oct/22

	$Great sir$
	Commented by aurpeyz last updated on 12/Oct/22

	$t h a n k y o u S i r . I g o t h o w y o u h a n d l e d t h e$ $f i r s t t w o b u t i d i d n t g w t h o w y o u d i d t h a t$ $o f t h e m u l t i p l e s o f 5 . p l s e x p l a i n$
	Commented by mr W last updated on 12/Oct/22

	${l e t}^{'} s b e g i n w i t h t h e l a s t d i g i t, w h i c h$ $s h o u l d n o t b e 5 . s o w e h a v e 4 w a y s t o$ $s e l e c t t h e l a s t d i g i t . f o r t h e s e c o n d$ $l a s t d i g i t w e h a v e a l s o 4 p o s s i b l i t i e s$ $a n d f o r t h e n e x t d i g i t 3 p o s s i b i l i t i e s$ $e t c . s o t a t a l l y w e h a v e 4 \times 4 \times 3 \times 2 \times 1$ $w a y s . a m o n g t h e m s o m e h a v e 1 a s$ $t h e f i r s t d i g i t . t h e s e n u m b e r s a r e$ $n o t v a l i d, b e c a u s e t h e y a r e s m a l l e r$ $t h a n 20000 . n o w w e s h o u d f i n d h o w$ $m a n g n u m b e r s b e g i n w i t h 1 .$ $1 a b c d$ $a b c d a r e f r o m 2, 3, 4, 5 . s i m i l a r y a s$ $a b o v e w e c a n g e t t h a t t h e r e a r e$ $3 \times 3 \times 2 \times 1 s u c h n u m b e r s . s o t h e$ $a n s w e r i s$ $4 \times 4 \times 3 \times 2 \times 1 - 3 \times 3 \times 2 \times 1 = 78 .$
	Commented by Acem last updated on 12/Oct/22

	$I a p p r e c i a t e t h e {p r o f e s s o r}^{'} s m e t h o d, {i t}^{'} s v e r y$ $c l e a r a n d e a s y . I w a s g o i n g t o m e n t i o n m i n e$ $b u t {i t}^{'} s a b i t c o m p l i c a t e d .$ $B i g t h a n k s$
	Commented by aurpeyz last updated on 12/Oct/22

	$t h a n k y o u s o m u c h$
	Commented by Acem last updated on 12/Oct/22

	$A s i m p l e e x p l a n a t i o n o f t h e a l m i g h t y$ ${p r o f e s s o r}^{'} s m e t h o d$ $a = 20 000$ $\begin{array}{cc} N o t 5 & {N W}_{{N u m}_{s} > a o r {N u m}_{s} < a} = 96 \\ 1 & 2 & 3 & 4 & 4 & \leftarrow S t a r t f r o m h e r e \end{array}$ $\begin{array}{cc} 1 & N o t 5 & {N W}_{{N u m}_{s} < a} = 18 \\ - & 1 & 2 & 3 & 3 & \leftarrow S t a r t f r o m h e r e \end{array}$ ${N u m I t e m s}_{H} = t h e d i f f e r e n c e = 78 n u m b e r s$
	Commented by aurpeyz last updated on 12/Oct/22

	$n i c e o n e$
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