Question

Find the sum of all numbers greater than $10000$ formed by using digits $0,2,4,6,8,$ no digit being repeated in any number.

Answer 1

To find required answer first we have to find the sum of all five-digit numbers and subtract the sum of numbers having first digit zero.

Total number of numbers formed by $0,2,4,6,8$ without repetition $=5!$

$=5\times 4\times 3\times 2\times 1$

$=120$

Sum of these numbers $=(5-1)!\times (0+2+4+6+8)\times (1+10+100+1000+10000)$

$=4!\times \times 20\times 11111$

$=53,33,280$

The numbers which include zero at beginnings, numbers of such number is $=4!$

$=4\times 3\times 2\times 1$

$=24$

Sum of the numbers having first digit zero $=(4-1)!\times (2+4+6+8)\times (1+10+100+1000)$

$=3!\times 20\times 1111$

$=1,33,320$

$\therefore$ The required answer $=53,33,280-1,33,320$

$=51,99,960$