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Question Number 199270 by mr W last updated on 31/Oct/23

Commented by mr W last updated on 31/Oct/23

$You can't use 'macro parameter character #' in math mode$
Commented by mr W last updated on 31/Oct/23

$i g o t 1111.1111 .$ $c a n s o m e o n e c o m f i r m ?$ $n i k i f 99 s i r : c a n y o u u s e y o u r f o r t r a n$ $p r o g r a m t o c h e c k i t ?$
Commented by nikif99 last updated on 31/Oct/23

$I g e t m e a n = 1234.43210678 .$ $e n d i n g d i g i t s i n b l u e m i g h t b e ‘ ‘ {n o i s e}^{″}$ $o f t h e p r o c e s s o r o r i n d e e d d e c i m a l$ $p l a c e s w h e n t o t a l d i v i d e d b y (5! \times 4) = 480 .$ $I w i l l c h e c k i t m o r e .$
Commented by mr W last updated on 31/Oct/23

$t h a n k s s i r s!$
	Answered by aleks041103 last updated on 31/Oct/23

	$l e t t h e n e e d e d a v e r a g e b e A$ $l e t t h e a v e r a g e o f a l l t h e n u m b e r s m a d e$ $w i t h o u t t h e d e c i m a l p o i n t b e B .$ $t h e n :$ $A = \frac{1}{4} (0.1 B + 0.01 B + 0.001 B + 0.0001 B) =$ $= \frac{0.1111}{4} B$ $f o r B :$ $B = \frac{1}{5!} \sum_{σ \in S_{5}} (σ (1) + 10 σ (2) + 10^{2} σ (4) + 10^{3} σ (5) + 10^{4} σ (8)) =$ $= \frac{4!}{5!} (1 + 10 + 10^{2} + 10^{3} + 10^{4}) (1 + 2 + 4 + 5 + 8) =$ $= \frac{11111}{5} 20 = 44444$ $\Rightarrow A = \frac{0.1111}{4} 44444$ $\Rightarrow A = 1234.4321$
	Commented by aleks041103 last updated on 31/Oct/23

	Commented by MathematicalUser2357 last updated on 04/Jan/24

	$CODING TIME!$
	Answered by mr W last updated on 31/Oct/23

	$t h i s i s m y s o l u t i o n :$ $g e n e r a l l l y, t h e s u m o f s o m e n u m b e r s$ $i s t h e s u m o f t h e v a l u e s w h i c h t h e i r$ $d i g i t s r e p r e s e n t . f o r e x a m p l e :$ $123 + 132 + 213 + 231 + 312 + 321 = 1332$ $b u t w e c a n a l s o g e t t h e s u m l i k e t h i s :$ $t h e d i g i t ‘ ‘ 1^{″} c a n s t a n d o n t h e f i r s t$ $p o s i t i o n w h i c h r e p r e s e n t s 100 . t h e r e$ $a r e 2! s u c h p o s s i b i l i t i e s . t h a t m e a n s$ $t h e s u m o f d i g i t ‘ ‘ 1^{″} o n t h e f i r s t$ $p o s i t i o n i s 1 \times 100 \times 2! . t h e d i g i t ‘ ‘ 1^{″}$ $c a n a l s o s t a n d o n t h e s e c o n d a n d$ $t h e t h i r d p o s i t i o n, s o t h e s u m w h i c h$ $t h e d i g i t ‘ ‘ 1^{″} r e p r e s e n t s i s$ $1 \times (100 + 10 + 1) \times 2! = 1 \times 111 \times 2!$ $a n d t h e s u m w h i c h a l l d i d i t s p r e s e n t$ $i s t h u s$ $(1 + 2 + 3) \times 111 \times 2! = 6 \times 111 \times 2 = 1332 .$ $n o w b a c k t o o u r q u e s t i o n .$ $w i t h t h e d i g i t s ‘ ‘ 1, 2, 4, 5, 8^{″} a n d t h e$ $d e c i m a l p o i n t ‘ ‘ .^{″} w e c a n f o r m t o t a l l y$ $5! \times 4 = 480 n u m b e r s . e a c h d i g i t ‘ ‘ 1^{″}$ $c a n r e p r e s e n t 1000, 100, 10, 1, 0.1,$ $0.01, 0.001, 0.0001 r e s p e c t i v e l y . t h e r e$ $a r e 4! p o s s i b i l i t i e s f o r t h e d i g i t ‘ ‘ 1^{″} o n$ $1000 s p o s i t i o n, 2 \times 4! p o s s i b i l i t i e s o n$ $100 s p o s i t i o n, 3 \times 4! o n 10 s p o s i t i o n,$ $4 \times 4! o n 1 s p o s i t i o n, 4 \times 4! o n 0.1 s$ $p o s i t i o n, 3 \times 4! o n 0.01 s p o s i t i o n,$ $2 \times 4! o n 0.001 p o s i t i o n a n d 1 \times 4!$ $o n 0.0001 p o s i t i o n . s o t h e s u m$ $w h i c h a l l d i g i t s r e p r e s e n t i s$ $(1 + 2 + 4 + 5 + 8) \times (1 \times 1000.0001 + 2 \times 100.001 + 3 \times 10.01 + 4 \times 1.1) \times 4!$ $s o t h e a v e r a g e o f t h e n u m b e r s i s$ $\frac{(1 + 2 + 4 + 5 + 8) \times (1 \times 1000.0001 + 2 \times 100.001 + 3 \times 10.01 + 4 \times 1.1) \times 4!}{5! \times 4}$ $= \frac{20 \times (1234.4321)}{5 \times 4}$ $= 1234.4321 ✓$
	Commented by aleks041103 last updated on 31/Oct/23

	$T h e p o b l e m i n y o u r s o l u t i o n c o m e s f r o m t h e f a c t$ $t h a t y o y a s s u m e t h e d i g i t c a n a l w a y s r e p r e s e n t$ $0.1, w h i c h i s i n c o r r e c t .$ $F o r e x a m p l e, i f 1 i s t h e f i r s t d i g i t, i t c a n n o t$ $h a v e a f a c t o r 0.1, b e c a u s e t h e n t h e$ $n u m b e r w o u l d b e 0.1 . . ., w h i c h i s n o t$ $a l l o w e d i n t h e s t a t e m e n t .$
	Commented by mr W last updated on 31/Oct/23

	$i {d i d n}^{'} t a s s u m e t h a t . i t j u s t m e a n t$ $t h a t ‘ ‘ 1^{″} c a n s t a n d f o r 0.1, f o r$ $e x a m p l e i n t h e n u m b e r s l i k e$ $2458.1, 4528.1, e t c .$
	Commented by aleks041103 last updated on 31/Oct/23

	$y e s . y o u a r e r i g h t .$ $a c t u a l l y t h e r e i s a n o t h e r p r o b l e m .$ $y o u a s s u m e t h e r e a r e 4! w a y s t o h a v e^{'} 1^{'}$ $a t t h e 10 s p o s i t i o n, f o r e x a m p l e .$ $t h i s i s t r u e i n t h e c a s e o f i n t e g e r s,$ $b u t t h e a d d i t i o n o f t h e d e c i m a l d o t$ $c h a n g e s t h i n g s .$ $e x a m p l e$ $1 i n 10 s p o s i t i o n$ $1) * * * 1 * .$ $2) * * 1 * . $ $3) 1 * . * $ $4) 1 . * * *$ $S o a c t u a l l y f o r t h e 10 s, t h e r e a r e n o t 4!$ $w a y s, b u t a c t u a l l y 4 \times 4! w a y s .$
	Commented by mr W last updated on 31/Oct/23

	$i g o t i t! t h a n k y o u s i r! i h a v e f i x e d$ $m y s o l u t i o n a n d g o t a l s o 1234.4321$ $n o w .$
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