How-many-odd-numbers-greater-than-60000-can-be-made-from-the-digits-5-6-7-8-9-0-if-no-number-contains-any-digit-more-than-ones-

Question Number 56688 by pieroo last updated on 21/Mar/19

How many odd numbers, greater than

60000, can be made from the digits

5, 6, 7, 8, 9, 0, if no number contains any

digit more than ones ?

Answered by mr W last updated on 22/Mar/19

6 X X X Y, 8 X X X Y : 2 \times 3 \times P_{3}^{4}

7 X X X Y, 9 X X X Y : 2 \times 2 \times P_{3}^{4}

Z X X X X Y : 3 \times 4 \times 4!

\Rightarrow (2 \times 3 + 2 \times 2) P_{3}^{4} + 3 \times 4 \times 4! = 528

Commented by malwaan last updated on 22/Mar/19

I d o n t u n d e r s t a n t

p l e a s e m o r e e x p l a i n a s i o n

Commented by rahul 19 last updated on 22/Mar/19

6 \underset{\underset{4 w a y s}{↓}}{-} \underset{\underset{3 w a y s}{↓}}{-} \underset{\underset{2 w a y s}{↓}}{-} \underset{\underset{^{3} C_{1}}{↓}}{-}

s i m i l a r l y f o r d i g i t s s t a r t i n g w i t h 8 .

7 \underset{\underset{4 w a y s}{↓}}{-} \underset{\underset{3 w a y s}{↓}}{-} \underset{\underset{2 w a y s}{↓}}{-} \underset{\underset{^{2} C_{1}^{}}{↓}}{-}

S i m i l a r l y f o r d i g i t s s t a r t i n g w i t h 9 .

\Rightarrow 2 (4 \times 3 \times 2 \times 3) + 2 (4 \times 3 \times 2 \times 2) = 240 .

(D e a r p i e r o o,

i n t h i s {s o l}^{n} i^{'} v e c o u n t e d 5 - d i g i t n o . s

o n l y! S o t h i s {s o l}^{n} i s i n c o m p l e t e .)

Commented by mr W last updated on 22/Mar/19

t h i s i s c o r r e c t f o r 5 - d i g i t n u m b e r s .

b u t y o u c a n a l s o m a k e 6 - d i g i t n u m b e r s .

t h e y a r e a l l g r e a t e r t h a n 60000 .

Commented by mr W last updated on 22/Mar/19

w i t h t h e 6 d i g i t s w e c a n m a k e 5 - d i g i t

n u m b e r s (l i k e 78065) a n d 6 - d i g i t

n u m b e r s (l i k e 567809) .

5 - d i g i t n u m b e r s g r e a t e r t h a n 6000 :

6 X X X Y : Y c a n b e 5, 7, 9 \Rightarrow 3 w a y s, X X X f r o m t h e r e m a i n i n g 4 d i g i t s \Rightarrow P_{3}^{4} w a y s, t o t a l l y 3 \times P_{3}^{4} w a y s .

8 X X X Y : a s 6 X X X Y, t o t a l l y 3 \times P_{3}^{4} w a y s .

7 X X X Y : Y c a n b e 5, 9 \Rightarrow 2 w a y s, X X X f r o m t h e r e m a i n i n g 4 d i g i t s \Rightarrow P_{3}^{4} w a y s, t o t a l l y 2 \times P_{3}^{4} w a y s .

8 X X X Y : a s 7 X X X Y, t o t a l l y 2 \times P_{3}^{4} w a y s .

\Rightarrow 5 - d i g i t n u m b e r s : 2 \times 3 \times P_{3}^{4} + 2 \times 2 \times P_{3}^{4} = 240

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

Z X X X X Y :

Y c a n b e 5, 7, 9 \Rightarrow 3 w a y s

Z c a n b e o n e o f t h e 4 r e m a i n i n g (0 n o t a l l o w e d) \Rightarrow 4 w a y s

X X X X f r o m t h e 4 r e m a i n i n g d i g i t s \Rightarrow 4! w a y s

\Rightarrow 6 - d i g i t n u m b e r s : 3 \times 4 \times 4! = 288

a l l p o s s i b l e n u m b e r s : 240 + 288 = 528

Commented by rahul 19 last updated on 22/Mar/19

T h a n k Y o u S i r! :)

Commented by malwaan last updated on 28/Mar/19

t h a n k y o u s o m u c h s i r

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