Find-the-number-of-five-digit-numbers-containing-exactly-three-different-digits-Examples-12312-12224-

Question Number 100444 by mr W last updated on 26/Jun/20

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

c o n t a i n i n g e x a c t l y t h r e e d i f f e r e n t

d i g i t s ? E x a m p l e s : 12312, 12224

Answered by Rasheed.Sindhi last updated on 28/Jun/20

^{∙} L e t a, b, c a r e d i g i t s d i f f e r e n t

f r o m e a c h o t h e r .

^{∙} F i r s t d i g i t (f r o m l e f t) : a

^{∙} F i r s t t w o d i g i t s : a \to a a a b

(2 n d d i g i t m a y b e s a m e o r d i f f e r e n t)

^{∙} F i r s t t h r e e d i g i t s :

a a \to a a a a a b

a b \to a b a a b b a b c

^{∙} F i r s t f o u r d i g i t s :

a a a \to a a a b

a a b \to a a b a a a b b a a b c

a b a \to a b a a a b a b a b a c

a b b \to a b b a a b b b a b b c

a b c \to a b c a a b c b a b c c

^{∙} F i v e d i g i t s (a l l p o s s i b l e c a s e s)

a a a b \to a a a b c

a a b a \to a a b a c

a a b b \to a a b b c

a a b c \to a a b c a a a b c b a a b c c

a b a a \to a b a a c

a b a b \to a b a b c

a b a c \to a b a c a a b a c b a b a c c

a b b a \to a b b a c

a b b b \to a b b b c

a b b c \to a b b c a a b b c b a b b c c

a b c a \to a b c a a a b c a b a b c a c

a b c b \to a b c b a a b c b b a b c b c

a b c c \to a b c c a a b c c b a b c c c

T o t a l p o s s i b l e c a s e s 25

★ I n e a c h c a s e :

(l e t U = {0, 1, \dots, 9})

^{∙} f i r s t a c a n b e f i l l e d i n

(U - {0}) 9 w a y s

^{∙} f i r s t b (f r o m l e f t) c a n b e f i l l e d

i n (U - {a}) 9 w a y s

^{∙} f i r s t c (f r o m l e f t) c a n b e f i l l e d

i n (U - {a, b}) 8 w a y s

^{∙} e a c h o f o t h e r a^{'} s, b^{'} s, c^{'} s c a n

b e f i l l e d i n 1 w a y

T o t a l w a y s i n e a c h c a s e

9 \times 9 \times 8 = 648

T o t a l o f 25 c a s e s :

648 \times 25 = 16200

Commented by mr W last updated on 04/Jul/20

t h a n k s f o r s o l v i n g s i r!

Answered by mr W last updated on 28/Jun/20

i f o u n d f o l l o w i n g g e n e r a l m e t h o d .

{l e t}^{'} s c o n s i d e r t h e g e n e r a l c a s e :

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

d i f f e r e n t d i g i t s (m ⩽ 10 ⩽ n) .

a t f i r s t {l e t}^{'} s t r e a t d i g i t 0 l i k e t h e o t h e r

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

w i t h 0 .

t o s e l e c t m d i g i t s f r o m t h e 10

a v a i l a b l e d i g i t s (0, 1, 2, \dots, 9) t h e r e a r e

C_{m}^{10} w a y s .

{l e t}^{'} s t r e a t t h e n d i g i t p o s i t i o n s i n a

n - d i g i t n u m b e r a s n b o x e s a n d w e

a r e g o i n g t o f i l l t h e s e b o x e s w i t h b a l l s

i n m d i f f e r e n t c o l o u r s a n d e a c h b o x

s h o u l d o b t a i n a t l e a s t o n e b a l l . t o d o

t h i s t h e r e a r e m! {_{m}^{n}} w a y s . {_{m}^{n}} a r e t h e

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

s e c o n d k i n d . {_{m}^{n}} i s t h e n u m b e r o f w a y s

t o p a r t i t i o n a s e t o f n o b j e c t s i n t o m

n o n - e m p t y s u b s e t s . f o r t h e m

c o l o u r s t o r e p r e s e n t t h e m d i f f e r e n t

d i g i t s t h e r e a r e m! w a y s . t h e r e f o r e

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

c a n b e f o r m e d b y e x a c t l y m d i f f e r e n t

d i g i t s i s C_{m}^{10} m! {_{m}^{n}} . b u t a m o n g t h e s e

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

b e g i n w i t h 0 . w e k n o w t h e r e a r e t h e

s a m e m a n y n u m b e r s b e g i n i n g w i t h 0

a s w i t h 1 o r w i t h 2 e t c . s o t h e n u m b e r

o f n u m b e r s w h i c h {d o n}^{'} t b e g i n w i t h 0

i s \frac{9}{10} \times C_{m}^{10} m! {_{m}^{n}} . t h i s i s t h e a n s w e r

o f o u r q u e s t i o n .

w i t h n = 5, m = 3 w e g e t

\frac{9}{10} \times C_{3}^{10} \times 3! \times {_{3}^{5}}

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

Commented by mr W last updated on 28/Jun/20

Commented by mr W last updated on 28/Jun/20

Commented by Rasheed.Sindhi last updated on 28/Jun/20

D e e p t h i n k i n g & N i c e r e s e a r c h!

Answered by ajfour last updated on 29/Jun/20

A n s w e r i s <^{10} C_{3} (5!) = \frac{10 \times 9 \times 8}{1 \times 2 \times 3} \times 120

\Rightarrow a n s w e r < 14, 400 (i t h i n k)

Commented by mr W last updated on 29/Jun/20

t h i s i s n o t c o r r e c t s i r .

w i t h 5 d i f f e r e n t d i g i t s y o u c a n f o r m

12345 \Rightarrow 5! = 120 n u m v e r s .

b u t w i t h t h r e e d i f f e r e n t d i g i t s 1, 2, 3

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

11123 : \Rightarrow \frac{5!}{3!}

12223 : \Rightarrow \frac{5!}{3!}

12333 : \Rightarrow \frac{5!}{3!}

11223 : \Rightarrow \frac{5!}{2! 2!}

11233 : \Rightarrow \frac{5!}{2! 2!}

12233 : \Rightarrow \frac{5!}{2! 2!}

\Rightarrow 3 \times \frac{5!}{3!} + 3 \times \frac{5!}{2! 2!} = 150 [= 3! {_{3}^{5}} = 6 \times 25] > 120

Commented by ajfour last updated on 02/Jul/20

t h a n k s f o r t h e l i g h t s i r, i s h a l l t r y

t o a r r i v e a t t h e a n s w e r, m y w a y t o o .

Answered by mr W last updated on 04/Jul/20

C l a s s i c a l w a y i a l s o u s e d :

t o s e l e c t 3 d i g i t s f r o m 10 t h e r e a r e

C_{3}^{10} = 120 w a y s .

t o b u i l d a f i v e d i g i t n u m b e r w i t h 3

d i f f e r e n t d i g i t s, s a y 1, 2, 3, t h e r e a r e

f o l l o w i n g c a s e s :

c a s e 1 : o n e d i g i t t r i p l e, t w o d i g i t s

o n c e e a c h :

11123, 22213, 33312 \Rightarrow 3 \times \frac{5!}{3!} w a y s

c a s e 2 : o n e d i g i t o n c e, t w o d i g i t s

t w i c e e a c h :

12233, 21133, 31122 \Rightarrow 3 \times \frac{5!}{2! 2!} w a y s

\Rightarrow t o t a l l y C_{3}^{10} \times 3 \times (\frac{5!}{3!} + \frac{5!}{2! 2!}) = 18000

b u t t h i s i n c l u d e s a l s o t h o s e n u m b e r s

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

t o b u i l d a n u m b e r b e g i n n i n g w i t h 0,

c a s e 1 : 01222, 02221 \Rightarrow 2 \times \frac{4!}{3!} \times C_{2}^{9}

c a s e 2 : 01122 \Rightarrow \frac{4!}{2! 2!} \times C_{2}^{9}

c a s e 3 : 00112, 00122 \Rightarrow 2 \times \frac{4!}{2!} \times C_{2}^{9}

c a s e 4 : 00012 \Rightarrow \frac{4!}{2!} \times C_{2}^{9}

\Rightarrow t o t a l l y (2 \times \frac{4!}{3!} + \frac{4!}{2! 2!} + 2 \times \frac{4!}{2!} + \frac{4!}{2!}) \times C_{2}^{9} = 1800

\Rightarrow t o t a l v a l i d f i v e - d i g i t n u m b e r s :

18000 - 1800 = 16200

Commented by Rasheed.Sindhi last updated on 04/Jul/20

N i c e a p p r o a c h : E a s y t o u n d e r s t a n d!