An-8-digit-number-divisible-by-9-is-to-be-formed-using-digits-from-0-to-9-without-repeating-the-digits-The-number-of-ways-in-which-this-can-be-done-is-a-72-7i-b-18-7i-c-40-7i-d-36-7i-please-I-m

Question Number 52276 by Necxx last updated on 05/Jan/19

A n 8 d i g i t n u m b e r d i v i s i b l e b y 9 i s

t o b e f o r m e d u s i n g d i g i t s f r o m

0 t o 9 w i t h o u t r e p e a t i n g t h e d i g i t s .

T h e n u m b e r o f w a y s i n w h i c h t h i s

c a n b e d o n e i s ?

a) 7 2 (7 i) b) 18 (7 i) c) 40 (7 i) d) 36 (7 i)

p l e a s e I m e a n f a c t o r i a l b y i . T h e

p a r t o f m y s c r e e n w h e r e t h e f a c t o r i a l

s y m b o l i s i s n t f u n c t i o n i n g . T h a n k s

Answered by mr W last updated on 06/Jan/19

w e h a v e 10 d i g i t s, b u t w e n e e d o n l y 8

f r o m t h e m t o f o r m a 8 - d i g i t n u m b e r .

t h a t i s t o s a y, w e m u s t t a k e 2 o f t h e

10 d i g i t s a w a y . b u t w h i c h d i g i t s s h a l l

w e t a k e a w a y ? i t i s r e q u e s t e d t h a t

t h e 8 - d i g i t n u m b e r s a r e d i v i s i b l e b y

9 . w e k n o w a n u m b e r i s d i v i s i b l e b y 9

o n l y w h e n t h e s u m o f i t s d i g i t s i s

d i v i s i b l e b y 9 . s i n c e 0 + 1 + 2 + \dots + 9 = 45,

i t m e a n s w h e n w e t a k e a l l 10 d i g i t s,

t h e n u m b e r s f o r m e d b y t h e m a r e a l w a y s

d i v i s i b l e b y 9, s i n c e 45 i s d i v i s i b l e b y 9 .

w h e n w e w a n t t o r e m o v e t w o d i g i t s t o

f o r m 8 - d i g i t n u m b e r s w h i c h r e m a i n

d i v i s i b l e b y 9, t h e s u m o f t h e s e t w o

d i g i t s m u s t b e d i v i s i b l e b y 9 . t h e r e a r e

o n l y f o l l o w i n g 5 p o s s i b i l i t i e s :

w e t a k e 0 a n d 9 a w a y,

w e t a k e 1 a n d 8 a w a y,

w e t a k e 2 a n d 7 a w a y,

w e t a k e 3 a n d 6 a w a y,

w e t a k e 4 a n d 5 a w a y .

w h e n w e t a k e 0 a n d 9 a w a y, w i t h t h e

r e m a i n i n g 8 d i g i t s 1 t o 8 w e c a n f o r m

t o t a l l y 8! n u m b e r s .

w h e n w e t a k e 1 a n d 8 a w a y, w i t h t h e

r e m a i n i n g 8 d i g i t s (o n e o f t h e m i s 0)

w e c a n f o r m t o t a l l y o n l y 7 \times 7! n u m b e r s

s i n c e 0 c a n n o t b e p l a c e d i n t h e f i r s t

p o s i t i o n .

s i n c e w e h a v e 4 s u c h s i m i l a r w a y s

t o t a k e t w o d i g i t s a w a y, w e c a n t o t a l l y

f o r m 4 \times 7 \times 7! n u m b e r s .

s o t h e r e s u l t i s 8! + 4 \times 7 \times 7! = 36 \times 7!

\Rightarrow a n s w e r d) i s c o r r e c t .

Commented by Necxx last updated on 06/Jan/19

w o w \dots . . T h a n k y o u s i r .

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