a-6-digit-number-formed-with-the-first-number-is-2-and-contains-4-equal-number-many-number-can-be-made-are-

Question Number 77394 by jagoll last updated on 06/Jan/20

a 6 digit number formed

with the first number is 2 and

contains 4 equal number . many

number can be made are

Commented by mr W last updated on 06/Jan/20

t h e f i r s t d i g i t i s 2, {t h a t}^{'} s c l e a r .

i t m u s t c o n t a i n 4 e q u a l d i g i t s . b u t

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

d i g i t s ? m a y t h e y t h e s a m e d i g i t o r

t h e y m u s t b e d i f f e r e n t d i g i t s ?

e x a m p l e : a r e 233332 a n d 233222

v a l i d ?

Commented by jagoll last updated on 06/Jan/20

v a l i d s i r

Commented by jagoll last updated on 06/Jan/20

i g o t 3500 s i r ? i t c o r r e c t ?

Commented by mr W last updated on 06/Jan/20

i t h i n k {i t}^{'} s n o t c o r r e c t s i r .

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

c a s e 1 :

i n t h e r e m a i n i n g 5 d i g i t s w e h a v e

3 t i m e s t h e d i g i t 2 .

c a s e 1 a : t y p e 2222 X X

\Rightarrow 9 \times \frac{5!}{3! 2!} = 90

c a s e 1 b : t y p e 2222 X Y

\Rightarrow C_{2}^{9} \times \frac{5!}{3!} = 720

c a s e 2 : t y p e 2 X X X X Y

i n t h e r e m a i n i n g 5 d i g i t s w e h a v e

4 t i m e s a n o t h e r d i g i t t h a n 2 .

\Rightarrow 9 \times 9 \times \frac{5!}{4!} = 405

t o t a l l y :

90 + 720 + 405 = 1215 n u m b e r s

Commented by jagoll last updated on 06/Jan/20

h o w a b o u t 2 X X X X 2 ?

Commented by jagoll last updated on 06/Jan/20

= 9 \times \frac{5!}{4!} = 4 5

Commented by mr W last updated on 06/Jan/20

t h i s i s i n c l u d e d i n 2 X X X X Y .

X c a n b e o n e o f 9 d i g i t s (0, 1, 3, 4, \dots 9)

Y c a n b e o n e o f 9 d i g i t s (0, 1, 2, 3, 4, \dots 9) - X

Commented by jagoll last updated on 06/Jan/20

o y e s s i r

Commented by jagoll last updated on 06/Jan/20

f i x t o t a l l y = 1215 s i r

Commented by peter frank last updated on 06/Jan/20

t h a n k y o u