you randomly select a 5 digit number. what′s the probability that this number has exactly 3 different digits?


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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
	$shorter way: 5 digit numbers with exact 3 different digits: (9/(10))×C_3 ^(10) ×3!× { (5),(3) :}} =(9/(10))×120×6×25=16200$
	$s h o r t e r w a y :$ $5 d i g i t n u m b e r s w i t h e x a c t 3 d i f f e r e n t$ $d i g i t s :$ $\frac{9}{10} \times C_{3}^{10} \times 3! \times {\begin{cases} 5 \\ 3 \end{cases}}$ $= \frac{9}{10} \times 120 \times 6 \times 25 = 16200$
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