Between-20000-and-70000-find-the-number-of-even-integers-in-which-no-digits-is-repeated-

Question Number 124709 by benjo_mathlover last updated on 05/Dec/20

B e t w e e n 20000 a n d 70000

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

i n w h i c h n o d i g i t s i s r e p e a t e d

Answered by liberty last updated on 05/Dec/20

L e t a b c d e b e a r e q u i r e d e v e n i n t e g e r .

A s s h o w n i n t h e f o l l o w i n g d i a g r a m, t h e

1^{s t} d i g i t c a n b e c h o s e n f r o m {2, 3, 4, 5, 6}

a n d t h e 5^{t h} d i g i t e c a n b e c h o s e n f r o m

{0, 2, 4, 6, 8} .

⌊ a ⌋ ⌊ b c d ⌋ ⌊ e ⌋

s i n c e {2, 3, 4, 5, 6} \cap {0, 2, 4, 6, 8} = {2, 4, 6}

w e d e v i d e t h e p r o b l e m i n t o 2 d i s j o i n t c a s e s

c a s e (1) a \in {2, 4, 6} t h e r e a r e = 3 \times 4 \times P_{3}^{8}

c a s e (2) a \in {3, 5} t h e r e a r e = 2 \times 5 \times P_{3}^{8}

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

i s 4032 + 3360 = 7392 .

Answered by mr W last updated on 06/Dec/20

A B C D E

A : 2, 3, 4, 5, 6

E : 0, 2, 4, 6, 8

c a s e 1 : E \in {0, 8}

\Rightarrow 2 \times 5 \times P_{3}^{8} = 3360

c a s e 2 : E \in {2, 4, 6}

\Rightarrow 3 \times 4 \times P_{3}^{8} = 4032

\Rightarrow t o t a l l y 3360 + 4032 = 7392

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