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Question Number 105791 by mr W last updated on 01/Aug/20

F r o m 1 t o 12345, h o w m a n y n u m b e r s

c o n t a i n t h e d i g i t 0 ? F i n d t h e n u m b e r

o f z e r o s i n a l l t h e s e n u m b e r s .

E x a m p l e : 10020 h a s t h r e e z e r o s .

Commented by PRITHWISH SEN 2 last updated on 01/Aug/20

189 + 2700 + 1630 = 4519

The answer is 4519 please check

Answered by mr W last updated on 02/Aug/20

N = n u m b e r o f n u m b e r s c o n t a i n i n g

a t l e a s t o n e z e r o

S = n u m b e r o f z e r o s i n t h e n u m b e r s

X, Y, Z = p l a c e h o l d e r n o n - z e r o d i g i t s

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

X \Rightarrow N = 0, S = 0

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

X 0 \Rightarrow N = 9, S = 9

t h r e e d i g i t n u m b e r s :

X 0 Y, X Y 0 \Rightarrow N = 2 \times 9^{2} = 162, S = 162

X 00 \Rightarrow N = 9, S = 2 \times 9 = 18

\Rightarrow t o t a l N = 162 + 9 = 171

\Rightarrow t o t a l S = 162 + 18 = 180

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

X 0 Y Z, X Y 0 Z, X Y Z 0 \Rightarrow N = 3 \times 9^{3} = 2187, S = 2187

X 00 Y, X Y 00, X 0 Y 0 \Rightarrow N = 3 \times 9^{2} = 243, S = 486

X 000 \Rightarrow N = 9, S = 27

\Rightarrow t o t a l N = 2187 + 243 + 9 = 2439

\Rightarrow t o t a l S = 2187 + 486 + 27 = 2700

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

10000 \Rightarrow N = 1, S = 4

1 X 000, 10 X 00, 100 X 0, 1000 X

\Rightarrow N = 2 + 9 + 9 + 9 = 29, S = 87

1 X Y 00, 10 X Y 0, 100 X Y, 1 X 0 Y 0, 1 X 00 Y, 10 X 0 Y

\Rightarrow N = (9 + 3) + 3 \times 9^{2} + 2 \times 2 \times 9 = 291

\Rightarrow S = 2 \times 291 = 582

10 X Y Z, 1 X 0 Y Z, 1 X Y 0 Z, 1 X Y Z 0

\Rightarrow N = 9^{3} + 2 \times 9^{2} + (9^{2} + 3 \times 9) + (9^{2} + 2 \times 9 + 4) = 1102

\Rightarrow S = 1102

\Rightarrow t o t a l N = 1 + 29 + 291 + 1102 = 1423

\Rightarrow t o t a l S = 4 + 87 + 582 + 1102 = 1775

t o t a l l y :

N = 9 + 171 + 2439 + 1423 = 4042

S = 9 + 180 + 2700 + 1775 = 4664

Commented by PRITHWISH SEN 2 last updated on 01/Aug/20

considering 2 zero of 5 digits

the no . of zero will be = 285 \times 2 = 570 instead of

285 \times 3 = 855 ∴ the total zero = 4519

please check sir .

Commented by mr W last updated on 01/Aug/20

i t w a s w r o n g f o r f i v e d i g i t n u m b e r s .

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

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