find the number of 5 digit natural numbers with strictly ascending digits whose sum is 20. example: 12458 is such a number


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Question Number 188362 by mr W last updated on 28/Feb/23

$f i n d t h e n u m b e r o f 5 d i g i t n a t u r a l$ $n u m b e r s w i t h s t r i c t l y a s c e n d i n g$ $d i g i t s w h o s e s u m i s 20 .$ $e x a m p l e : 12458 i s s u c h a n u m b e r$
	Answered by ARUNG_Brandon_MBU last updated on 28/Feb/23

	$6$
	Commented by ARUNG_Brandon_MBU last updated on 28/Feb/23
	#include <iostream> using namespace std; int main(void) { for (int i0, i1, i2, i3, i4, i=10000; i<100000; i++) { i0 = i/10000; i1 = i%10000/1000; i2 = i%1000/100; i3 = i%100/10; i4 = i%10; if (i0<i1 && i1<i2 && i2<i3 && i3<i4) if ((i0+i1+i2+i3+i4) == 20) cout << i <<", "; } return 0; }
	Commented by ARUNG_Brandon_MBU last updated on 28/Feb/23

	$Output : 12359, 12368, 12458,$ $12467, 13457, 23456,$
	Commented by mr W last updated on 28/Feb/23

	$y e s!$
	Commented by ARUNG_Brandon_MBU last updated on 28/Feb/23
	��
	Commented by mr W last updated on 01/Mar/23

	$i n t h i s c a s e, i m e a n w h e n t h e s u m o f$ $d i g i t s i s ⩽ 20, w e c a n s t i l l h a v e a$ $s m a r t s o l u t i o n, w h i c h g i v e s u s t h e$ $a n s w e r 7 - 1 = 6 .$ $7 i s t h e c o e f . o f t e r m x^{20} i n t h e$ $e x p a n s i o n \underset{k = 1}{\prod^{5}} \frac{x^{k}}{1 - x^{k}} . b u t i t i n c l u d e s$ $a l s o t h e c a s e t h a t a d i g i t i s 10, w h i c h$ $i s c e r t a i n l y n o t t r u e, t h e r e f o r e t h e$ $c o r r e c t a n s w e r i s 7 - 1 = 6 .$ $i f t h e s u m o f d i g i t s i s > 20, i t h i n k$ $w e c a n o n l y e n u m e r a t e a l l p o s s i b l e$ $p o s s i b i l i t i e s a s y o u r p r o g r a m d o e s .$ $t h e r e i s n o s m a r t e r a p p r o a c h .$
	Commented by ARUNG_Brandon_MBU last updated on 28/Feb/23
	��
	Commented by mr W last updated on 04/Mar/23

	$i f o u n d a n e w w a y e v e n f o r n > 20,$ $s e e b e l o w . c a n y o u p l e a s e c h e c k w i t h$ $y o u r p r o g r a m ?$
	Commented by ARUNG_Brandon_MBU last updated on 07/Mar/23
	// for n > 20 #include <iostream> using namespace std; int main(void) { for(int i0,i1,i2,i3,i4,i=10000;i<100000;i++) { i0 = i/10000; i1 = i%10000/1000; i2 = i%1000/100; i3 = i%100/10; i4 = i%10; if (i0<i1 && i1<i2 && i2<i3 && i3<i4) if (i0+i1+i2+i3+i4 > 20) cout << i <<" "; } return 0; }
	Commented by ARUNG_Brandon_MBU last updated on 07/Mar/23

	Commented by ARUNG_Brandon_MBU last updated on 07/Mar/23

	$We have 108 5 - digit numbers such that$ $the digits are different and their sum > 20$
	Commented by mr W last updated on 07/Mar/23

	$y e s, {t h a t}^{'} s c o r r e c t!$ $a s w e c a n s e e b e l o w t h e r e a r e$ $8 n u m b e r s w i t h s u m 21,$ $9 n u m b e r s w i t h s u m 22,$ $11 n u m b e r s w i t h s u m 23,$ $. . .$ $2 n u m b e r s w i t h s u m 33,$ $1 n u m b e r w i t h s u m 34,$ $1 n u m b e r w i t h s u m 35 .$ $s o t o t a l l y 108 n u m b e r s w i t h s u m > 20 .$
	Answered by mr W last updated on 04/Mar/23

	$a n e w t r y$ $s a y t h e n u m b e r i s d_{1} d_{2} d_{3} d_{4} d_{5} w i t h$ $d_{1} + d_{2} + d_{3} + d_{4} + d_{5} = n a n d$ $1 ⩽ d_{1} < d_{2} < d_{3} < d_{4} < d_{5} ⩽ 9$ $l e t$ $d_{1} = 1 + k_{1} w i t h 0 ⩽ k_{1} ⩽ 8$ $d_{2} = d_{1} + 1 + k_{2} = 2 + k_{1} + k_{2} w i t h 0 ⩽ k_{2} ⩽ 7$ $d_{3} = d_{2} + 1 + k_{3} = 3 + k_{1} + k_{2} + k_{3} w i t h 0 ⩽ k_{3} ⩽ 6$ $. . . . . .$ $d_{5} = d_{4} + 1 + k_{4} = 5 + k_{1} + k_{2} + k_{3} + . . . + k_{5} w i t h 0 ⩽ k_{3} ⩽ 4$ $(1 + 2 + 3 + . . . + 5) + 5 k_{1} + 4 k_{2} + 3 k_{3} + 2 k_{4} + k_{5} = n$ $n u m b e r o f s o l u t i o n s i s t h e c o e f . o f$ $t e r m x^{n} i n t h e e x p a n s i o n o f$ $x^{15} \underset{r = 1}{\prod^{5}} \frac{1 - x^{10 - r}}{1 - x^{r}}$ $f o r n = 20 w e h a v e 6 n u m b e r s a n d$ $f o r n = 25 w e h a v e 12 n u m b e r s e t c .$
	Commented by mr W last updated on 04/Mar/23

	Commented by mr W last updated on 04/Mar/23

	$i n c a s e o f 6 d i g i t n u m b e r s w e n e e d$ $t o f i n d t h e c o e f . o f t e r m x^{n} i n$ $x^{21} \underset{r = 1}{\prod^{6}} \frac{1 - x^{10 - r}}{1 - x^{r}}$ $f o r n = 25 w e h a v e 4 n u m b e r s a n d$ $f o r n = 30 w e h a v e 8 n u m b e r s e t c .$
	Commented by mr W last updated on 04/Mar/23

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