Question

Find the number of odd integers between 1000 and 8000 which have none of their digits repeated.

Answer 1

There are $two$ restrictions operating this question.

$1.$ For a number to be odd, the unit digit should be either $1,3,5,7$ or $9.$

$2.$ $Thousand$ can not be filled with $8$ or $9.$ For unit digit when it is filled with $9$, the thousands place can be filled in $7$ ways.

Namely any digit from 1 to 7 the remaining $two$ places can be filled in $8\times 7=56$ ways.

$\therefore \text{Total no. formed in this way}=56\times 7=392$

Now if units place is filled with any of the four digits $1,3,5$ or $7,$ the $thousan{d}^{\prime }s$ place can be filled in $6$ ways and the remaining $two$ places can be filled in $8\times 7=56$ Ways.

$\therefore \text{The total no. formed in the way}=56\times 6\times 4=1344.$

$\therefore \text{Total}=392+1344=1736$