find-the-number-of-numbers-less-than-10-6-which-contain-at-least-3-different-digits-

Question Number 180421 by mr W last updated on 12/Nov/22

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

10^{6} w h i c h c o n t a i n a t l e a s t 3 d i f f e r e n t

d i g i t s .

Answered by liuxinnan last updated on 12/Nov/22

m a y b e 89965

Commented by liuxinnan last updated on 12/Nov/22

Commented by mr W last updated on 13/Nov/22

t h a n k s f o r t r y i n g s i r!

Answered by mr W last updated on 13/Nov/22

f r o m 100 t o 999999 \Rightarrow 999900 n u m b e r s

n u m b e r s w i t h e x a c t l y o n e d i g i t :

x x x, x x x x, x x x x x, x x x x x x

\Rightarrow 4 \times 9 = 36

n u m b e r s w i t h e x a c t l y t w o d i g i t s :

x x y, x x x y, x x x x y, x x x x x y

\Rightarrow (3 + 4 + 5 + 6) \times 9 \times 9 = 1458

999900 - 36 - 1458 = 998406 n u m b e r s

h a v e a t l e a s t t h r e e d i g i t s .

Commented by Acem last updated on 13/Nov/22

S i r, a t l e a s t 3 d i f f d i g = m e a n s t h r e e o r f o u r

o r s i x . e x a m p l e s 765, 2365,

22345, 12345

333678, 339654, 123456

I f t h a t w a s {w h a t}^{'} s m e a n t t h e n

I g o t 178 884

I f n o t, p l e a s e l e t m e u n d e r s t a n d t h e q u e s t i o n

Commented by mr W last updated on 13/Nov/22

a t l e a s t t h r e e \equiv t h r e e o r f o u r o r f i v e

o r s i x d i f f e r e n t d i g i t s a r e i n e a c h

n u m b e r .

e . g . 123123, 123412, 123451, 123456

a r e o k, b u t 122222, 333333 n o t o k .

12312, 12341, 12345 a r e o k, b u t 12112,

11111 a r e n o t o k .

1231, 1234 a r e o k, b u t 1212, 2222 n o t .

123 i s o k, b u t 121, 333 a r e n o t o k .

e t c .

Commented by Acem last updated on 14/Nov/22

T h a n k y o u S i r!, t h e p r o b l e m i s a n d f o r e x a m p l e

t h a t y o u d e e m t h a t t h e n u m b e r 123451 a s

5 d i f f . d i g . w h i l e i d o i t a s 4 d i f f . o n e w i t h 2 s a m e

n u m b e r s “ 1^{″} t h a t i s w h y i {d i d n}^{'} t c o m p t 5 d i f f .

T h e 2 n d e x a m p l e 123123 i d e e m i t a s t h r e e

d i f f . p a i r s 11, 22, 33 e t c a n d i d e e m t h e 3 d i f f . a s

o n l y X X X a b c w i t h i t p e r m u t a t i o n; x \neq a, b n o r c

A n d i a g r e e w i t h y o u i n r e m a i n c a s e s

T h a n k y o u b r o!

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