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Question Number 188551 by mr W last updated on 03/Mar/23

y o u r a n d o m l y s e l e c t a 5 d i g i t n u m b e r .

{w h a t}^{'} s t h e p r o b a b i l i t y t h a t t h i s n u m b e r

h a s e x a c t l y 3 d i f f e r e n t d i g i t s ?

Answered by mr W last updated on 03/Mar/23

f r o m 10000 t o 99999 t h e r e a r e t o t a l l y

90000 5 d i g i t n u m b e r s .

n o w w e w a n t t o f i n d h o w m a n y a m o n g

t h e m h a v e e x a c t l y 3 d i g i t s .

t o s e l e c t 3 d i g i t s f r o m 0 t o 9 t h e r e a r e

C_{3}^{10} p o s s i b i l i t i e s . s a y a, b, c a r e t h e

d i g i t s w e h a v e s e l e c t e d . t o f o r m a 5

d i g i t n u m b e r u s i n g t h e m w e h a v e

f o l l o w i n g p o s s i b i l i t i e s :

1) a a a b c

\Rightarrow C_{1}^{3} \times \frac{5!}{3!} = 60 n u m b e r s

2) a a b b c

\Rightarrow C_{1}^{3} \times \frac{5!}{2! 2!} = 90 n u m b e r s

i n s u m w e h a v e 150 n u m b e r s u s i n g

a, b, c

\Rightarrow 150 \times C_{3}^{10} = 18000 n u m b e r s

b u t a m o n g t h e m s o m e b e g i n w i t h t h e

d i g i t 0 l i k e 0 a b a b .

t o s e l e c t a, b f r o m 1 t o 9 t h e r e a r e

C_{2}^{9} p o s s i b i l i t i e s .

1) 0 a a a b

\Rightarrow 2 \times \frac{4!}{3!} = 8 n u m b e r s

2) 0 a a b b

\Rightarrow \frac{4!}{2! 2!} = 6 n u m b e r s

3) 000 a b

\Rightarrow \frac{4!}{2!} = 12 n u m b e r s

4) 00 a a b

\Rightarrow 2 \times \frac{4!}{2!} = 24 n u m b e r s

i n s u m t h e r e a r e

(8 + 6 + 12 + 24) \times C_{2}^{9} = 1800 n u m b e r s

w h i c h b e g i n w i t h 0 .

s o w e h a v e 18000 - 1800 = 16200

n u m b e r s w i t h e x a c t l y 3 d i g i t s .

p r o b a b i l i t y p = \frac{16200}{90000} = 18 % ✓

Commented by mr W last updated on 03/Mar/23

s h o r t e r w a y :

5 d i g i t n u m b e r s w i t h e x a c t 3 d i f f e r e n t

d i g i t s :

\frac{9}{10} \times C_{3}^{10} \times 3! \times {\begin{cases} 5 \\ 3 \end{cases}}

= \frac{9}{10} \times 120 \times 6 \times 25 = 16200

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