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What is the total number of positive integers that can be formed using the digits 1 , 2 , 3 , 4 , 3 , 2 , 1 1,2,3,4,3,2,1 when the odd digits are in odd place?


The answer is 18.

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2 solutions

Rohan Karmakar
Jun 16, 2014

2 , 4 , 2 , c a n b e a r r e n g e d i n t h r e e e v e n p l a c e s i n 3 ! 2 ! = 3 w a y s a n d 1 , 3 , 3 , 1 c a n b e a r r e n g e d i n f o u r o d d p l a c e s i n 4 ! 2 ! 2 ! = 6 w a y s . S o r e q u i r e d n u m b e r o f n u m b e r s i s 3 × 6 = 18 2,4,2,\quad can\quad be\quad arrenged\quad in\quad three\quad even\quad places\quad in\frac { 3! }{ 2! } =3\quad ways\\ \quad and\quad 1,3,3,1\quad can\quad be\quad arrenged\quad in\quad four\quad odd\quad places\quad in\quad \frac { 4! }{ 2!2! } =6\quad ways.\\ \quad So\quad required\quad number\quad of\quad numbers\quad is\quad 3\times 6=18

Here are the 6 6 possible odd places of the seven digit number

1 _ 3 _ 3 _ 1

1 _ 3 _ 1 _ 3

1 _ 1 _ 3 _ 3

3 _ 1 _ 1 _ 3

3 _ 1 _ 3 _ 1

3_ 3 _ 1 _ 1

Here are the 3 3 possible combinations of even places for every combination above of odd places

_ 2 _ 4 _ 2 _

_ 2 _ 2 _ 4 _

_ 4 _ 2 _ 2 _

Giving us 3 6 = 18 3*6 = \boxed{18} total possibilities.

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