How-many-5-digit-numbers-with-different-digits-are-multiple-of-9-

Question Number 178395 by mr W last updated on 16/Oct/22

H o w m a n y 5 d i g i t n u m b e r s w i t h

d i f f e r e n t d i g i t s a r e m u l t i p l e o f 9 ?

Answered by cortano1 last updated on 16/Oct/22

(1) 1 + 5 + 6 + 7 + 8 \Rightarrow 5!

(2) 1 + 2 + 3 + 4 + 8 \Rightarrow 5!

(3) 1 + 2 + 4 + 5 + 6 \Rightarrow 5!

(4) 1 + 2 + 3 + 5 + 7 \Rightarrow 5!

(5) 1 + 3 + 6 + 8 + 9 \Rightarrow 5!

(6) 1 + 4 + 5 + 8 + 9 \Rightarrow 5!

(7) 1 + 2 + 7 + 8 + 9 \Rightarrow 5!

(8) 2 + 3 + 5 + 8 + 9 \Rightarrow 5!

\Rightarrow 8 \times 5! = 8 \times 120 = 960

Commented by mr W last updated on 16/Oct/22

w h a t a b o u t n u m b e r s l i k e 13068, 47981

e t c . ?

Commented by Tawa11 last updated on 16/Oct/22

Great sir

Commented by mr W last updated on 16/Oct/22

i m e a n t i n h i s s o l u t i o n n u m b e r s l i k e

13068, 47981 e t c . a r e n o t i n c l u d e d .

Commented by mr W last updated on 16/Oct/22

s o a n s w e r 960 i s n o t c o r r e c t .

Commented by cortano1 last updated on 16/Oct/22

do you meant the numbers is

formed from 0, 1, \dots, 9 ?

Commented by mr W last updated on 16/Oct/22

y e s . a l l d i g i t s a r e a l l o w e d .

Commented by cortano1 last updated on 16/Oct/22

ok sir

Commented by mr W last updated on 16/Oct/22

t y p o . {i t}^{'} s j u s t a n e x a m p l e .

Commented by cortano1 last updated on 16/Oct/22

what your answer sir ?

Commented by mr W last updated on 16/Oct/22

i g o t 3024 .

Answered by mr W last updated on 16/Oct/22

0 + 1 + 2 + 3 + 4 = 10

9 + 8 + 7 + 6 + 5 = 35

t h a t m e a n s t h e s u m o f 5 d i f f e r e n t

d i g i t s l i e s b e t w e e n 10 a n d 35 .

s u c h t h a t a 5 d i g i t n u m b e r i s

m u l t i p l e o f 9, t h e s u m o f i t s d i g i t s

m u s t b e 27 o r 18 .

c a s e 1 : s u m o f d i g i t s = 27

9 + 8 + 7 + 3 + 0 \Rightarrow \frac{4}{5} \times 5! n u m b e r s

9 + 8 + 6 + 4 + 0

9 + 7 + 6 + 5 + 0

9 + 8 + 7 + 2 + 1 \Rightarrow 5! n u m b e r s

9 + 8 + 6 + 3 + 1

9 + 8 + 5 + 4 + 1

9 + 8 + 5 + 3 + 2

9 + 7 + 6 + 4 + 1

9 + 7 + 6 + 3 + 2

9 + 7 + 5 + 4 + 2

9 + 6 + 5 + 4 + 3

8 + 7 + 6 + 5 + 1

8 + 7 + 6 + 4 + 2

8 + 7 + 5 + 4 + 3

\Rightarrow 3 \times \frac{4}{5} \times 5! + 11 \times 5!

c a s e 2 : s u m o f d i g i t s = 18

9 + 6 + 2 + 1 + 0 \Rightarrow \frac{4}{5} \times 5! n u m b e r s

9 + 5 + 3 + 1 + 0

9 + 4 + 3 + 2 + 0

8 + 7 + 2 + 1 + 0

8 + 6 + 3 + 1 + 0

8 + 5 + 4 + 1 + 0

8 + 5 + 3 + 2 + 0

7 + 6 + 4 + 1 + 0

7 + 6 + 3 + 2 + 0

7 + 5 + 4 + 2 + 0

6 + 5 + 4 + 3 + 0

8 + 4 + 3 + 2 + 1 \Rightarrow 5! n u m b e r s

7 + 5 + 3 + 2 + 1

6 + 5 + 4 + 2 + 1

\Rightarrow 11 \times \frac{4}{5} \times 5! + 3 \times 5!

t o t a l l y :

\Rightarrow (11 + 3) (1 + \frac{4}{5}) \times 5! = 14 \times 9 \times 4! = 3024

Commented by SLVR last updated on 25/Oct/22

i t i s m o r e c l e a r \dots a w e s o m . s u p e r

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