A-certain-personal-identity-number-pin-is-a-sequence-of-four-digits-where-each-digit-is-selected-from-the-digits-0-to-9-How-many-such-pins-have-at-least-two-diffrent-digits-and-are-palindromes-Pa

Question Number 177633 by aurpeyz last updated on 07/Oct/22

A certain personal identity number (pin)

is a sequence of four digits where each digit

is selected from the digits 0 to 9 . How many

such pins have at least two diffrent digits

and are palindromes . (Palindrome is a cha -

racter that reads the same forward and

backward e . g 1221)

Commented by Strengthenchen last updated on 07/Oct/22

s u m o f k i n d s c o n b i n a t i o n i s 10 \times 10 \times 10 \times 10 = 10000

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

[p r o v e : i f t h e r e h a v e 1 e l e m e n t s i n 4 s q u e n c e, o n l y h a v e 1 s o l u t i o n

2 4 2 \times 2 \times 2 \times 2 = 16 [16]

3 4 3 \times 3 \times 3 \times 3 = 81 [81]

4 4 4 \times 4 \times 4 \times 4 = 264 [255]

5 4 625

p r o v e f i n i s h e d .]

s i m p l y i f c h o o s e o n e n u m b e r i n t h e f i r s t s p o t, i n t h e s e c o n d o n l y h a v e

9 n u m b e r i s d i f f e r e n t, t h e t h i r d s p o t h a v e 8, t h u s,

t h e r e h a v e 10 \times 9 \times 8 = 720 . {d o n}^{'} t h a v e d i f f e r e n t n u m b e r .

10000 - 720 = 9230 k i n d s t h i n g s h a s s a m e n u m b e r

a n d i f w a n t t o n e r r o w m u m b e r i s s e a m, i f t h e f i r s t s p o t

i s a, t h e n a a 00 - a a 99 h a v e 99 . a s t h i s b a a 1 - b a a 9 ? m i n u s

f i r s t s p o t s o l u t i o n s, h a v e 9 k i n d s, s o o n g e t 10 \times 100 + 9 \times 10 + 8 \times 1 = 1098 k i n d s

.

Commented by mr W last updated on 07/Oct/22

{i t}^{'} s w r o n g s i r .

t h e p i n s s h o u l d b e p a l i n d r o m e s, i . e .

i n f o r m o f a b b a . s o w e o n l y n e e d t o

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

g e t . {i t}^{'} s e a s y t o s e e w e h a v e 90 s u c h

n u m b e r s, n a m e l y :

a b = 10, 11, 12, \dots, 98, 99 .

t o t a l l y 90 n u m b e r s . s o t h e a n s w e r i s

90 .

({i t}^{'} s a s s u m e d t h a t t h e n u m b e r s m a y

n o t b e g i n w i t h 0)

Commented by Strengthenchen last updated on 08/Oct/22

o o o o p s! L e a r n i n g e n g l i s h w e l l i s i m p o r t a n t!

Commented by Tawa11 last updated on 08/Oct/22

Great sirs

Answered by mr W last updated on 07/Oct/22

X X X X \Rightarrow 9 n u m b e r s (0000 e x c l u d e d)

X Y Y X \Rightarrow 9 \times 9 = 81 n u m b e r s (0 Y Y 0 e x c l u d e d)

\Rightarrow t o t a l l y 9 + 81 = 90 p i n s

Commented by mr W last updated on 08/Oct/22

i n s e n s e o f t h e q u e s t i o n y o u a r e r i g h t

s i r . t h a n k s!

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