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


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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

	$t h i s i s i n c l u d e d i n 2 X X X X Y .$ $X c a n b e o n e o f 9 d i g i t s (0, 1, 3, 4, . . . 9)$ $Y c a n b e o n e o f 9 d i g i t s (0, 1, 2, 3, 4, . . . 9) - X$
	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

	$t h a n k y o u$
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