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Question Number 51287 by aseerimad last updated on 25/Dec/18

	Answered by mr W last updated on 26/Dec/18

	${l e t}^{'} s l o o k a t a n e x a m p l e :$ $t h e n u m b e r 1234 c o n s i s t s o f 4 d i g i t s :$ $1 r e p r e s e n t s h e r e t h e v a l u e 1 \times 1000$ $2 r e p r e s e n t s h e r e t h e v a l u e 2 \times 100$ $3 r e p r e s e n t s h e r e t h e v a l u e 3 \times 10$ $4 r e p r e s e n t s h e r e t h e v a l u e 4 \times 1$ $\Rightarrow 1234 = 1 \times 1000 + 2 \times 100 + 3 \times 10 + 4 \times 1$ $s o w h e n w e w a n t t o g e t t h e s u m o f s o m e$ $n u m b e r s, w e o n l y n e e d t o c a l c u l a t e$ $t h e s u m o f t h e v a l u e s w h i c h t h e i r$ $d i g i t s r e p r e s e n t . f o r e x a m p l e :$ $1234 + 4321$ $= (1 + 4) \times 1000 + (2 + 3) \times 100 + (3 + 2) \times 10 + (4 + 1) \times 1$ $= 5 \times (1000 + 100 + 10 + 1)$ $= 5 \times 1111$ $= 5555$ $w e c a n u s e t h i s m e t h o d t o s o l v e o u r$ $p r o b l e m . i t s a d v a n t a g e i s t h a t w e$ ${d o n}^{'} t n e e d t o k n o w t h e n u m b e r s$ $t h e m s e l f . w e o n l y n e e d t o k n o w t h e$ $d i g i t s f r o m w h i c h t h e n u m b e r s a r e$ $f o r m e d . a n d w e k n o w t h e d i g i t s, t h e y$ $a r e 1, 2, 3 a n d 4 .$ ${l e t}^{'} s l o o k a t t h e d i g i t 1 :$ $i t c a n b e a t t h e t h o u s a n d s p l a c e a s$ $1 X Y Z . w e k n o w t h e r e a r e 3! n u m b e r s$ $i n w h i c h t h e d i g i t 1 i s a t t h e t h o u s a n d s$ $p l a c e . t h e s u m o f a l l v a l u e s w h i c h t h e$ $d i g i t 1 r e p r e s e n t i n t h e t h o u s a n d s p l a c e$ $i s t h e n 1 \times 1000 \times 3! .$ $s i m i l a r l y t h e d i g i t 1 c a n b e i n t h e$ $h u n d r e d s p l a c e . t h e r e a r e a l s o 3! s u c h$ $n u m b e r s . t h e s u m o f a l l v a l u e s w h i c h t h e$ $d i g i t 1 r e p r e s e n t i n t h e h u n d r e d s p l a c e$ $i s 1 \times 100 \times 3! .$ $s i m i l a r l y t h e d i g i t 1 c a n b e i n t h e$ $t e n s p l a c e . t h e r e a r e a l s o 3! s u c h$ $n u m b e r s . t h e s u m o f a l l v a l u e s w h i c h t h e$ $d i g i t 1 r e p r e s e n t i n t h e t e n s p l a c e$ $i s 1 \times 10 \times 3! .$ $f i n a l l y t h e d i g i t 1 c a n b e i n t h e$ $u n i t s p l a c e . t h e r e a r e a l s o 3! s u c h$ $n u m b e r s . t h e s u m o f a l l v a l u e s w h i c h t h e$ $d i g i t 1 r e p r e s e n t i n t h e u n i t s p l a c e$ $i s 1 \times 1 \times 3! .$ $i n t h i s w a y w e g e t t h e s u m o f a l l$ $v a l u e s w h i c h t h e d i g i t 1 r e p r e s e n t i n$ $a l l p o s s i b l e p l a c e s i s :$ $1 \times (1000 + 100 + 10 + 1) \times 3!$ $t h i s a p p l i e s a l s o f o r t h e d i g i t s 2, 3, 4 .$ $t h e s u m o f a l l v a l u e s w h i c h a l l o f$ $t h e s e 4 d i g i t s i n a l l p o s s i b l e p l a c e s$ $r e p r e s e n t i s t h e n$ $(1 + 2 + 3 + 4) \times (1000 + 100 + 10 + 1) \times 3!$ $t h i s i s a l s o t h e s u m o f a l l n u m b e r s$ $f o r m e d b y t h e s e 4 d i g i t s .$ $s o t h e a n s w e r f o r t h e q u e s t i o n i s$ $(1 + 2 + 3 + 4) \times (1000 + 100 + 10 + 1) \times 3!$ $= 10 \times (1000 + 100 + 10 + 1) \times 3!$ $= 10 \times 1111 \times 3!$ $= 10 \times 6666$ $= 66660$
	Commented by aseerimad last updated on 26/Dec/18

	$c a n y o u p l s e x p l a i n t h e m e t h o d ?$
	Commented by mr W last updated on 26/Dec/18

	$i h a v e a d d e d s o m e e x p l a n a t i o n i n t o$ $t h e w o r k i n g s . h o p e {i t}^{'} s c l e a r n o w .$
	Commented by aseerimad last updated on 26/Dec/18

	$T h a n k y o u T h a n k y o u v e r y m u c h . T h a n k y o u f o r e x p l a i n i n g i n s u c h a l u c i d w a y .$ $G o d B l e s s y o u .$
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