How many numbers can be formed with the digits 2, 3, 4, 5, 4, 3, 2 so that the odd digits occupy the odd places?
The table shows the places where the odd digits can be placed
Odd | odd | odd | Odd |
There are 4 places
And 3 odd digits out of which 2 are of the same kind
Choose any 3 places out of the four places in 4C3 ways = 4 ways
In each way, the 3 digits can be placed in 
ways = 3 ways
total number of ways in which odd digits occupy odd places = 
= 12
Now there are 4 remaining digits out of which 2 are same of 1 kind, and 2 are same as another kind
they can be arranged in the remaining places in 
= 6 ways
total number of numbers where odd digit occupies odd places = 12 × 6 = 72
There are 72 such numbers
1