Question

The sum of all numbers greater than 10000 by using the digits $0,2,4,6,8$ considering no digit being repeated in any number is:
A. $5199970$
B. $5199960$
C. $5199950$
D. $5199940$

Answer 1

Hint: In the given question first we will make an equation for the sum of numbers including zero it means for a total of five digits then we will make an equation for the sum of numbers excluding zero it means for total four digits. After that for the condition that no digit being repeated in any number we will take the difference of both. Thus we will get the answer.

Complete step by step Answer:

From the given question:
Total numbers formed by $0,2,4,6,8$ without repetition is: $5!$
Now we solve it:
We get:
$
  5! = 5 \times 4 \times 3 \times 2 \times 1 \\
   \Rightarrow 120 \\
 $
Now:
Sum of these numbers:
$
  \left( {5 - 1} \right)! \times \left( {0 + 2 + 4 + 6 + 8} \right) \times 1111 \\
   \Rightarrow 4! \times 20 \times 1111 \\
   \Rightarrow 4 \times 3 \times 2 \times 1 \times 20 \times 1111 \\
   \Rightarrow 5333280 \\
 $
This would include zero at the beginning, such number is$ = 4!$
$
   \Rightarrow 4 \times 3 \times 2 \times 1 \\
   \Rightarrow 24 \\
 $
Sum of such type of numbers:
$
   = \left( {4 - 1} \right)! \times \left( {2 + 4 + 6 + 8} \right) \times 1111 \\
   \Rightarrow 3! \times 20 \times 1111 \\
   \Rightarrow 3 \times 2 \times 1 \times 20 \times 1111 \\
   \Rightarrow 133320 \\
 $
Now we will take the difference between both conditions:
We get:
$
   \Rightarrow 5333280 - 133320 \\
   \Rightarrow 5199960 \\
 $
And this is our answer.
Thus the sum of all numbers greater than 10000 by using the digits $0,2,4,6,8$ considering no digit being repeated in any number is $5199960$.
Hence the correct answer is option B.

Note: In the given question remember that we have to make two conditions, first is for including zero number and second is for excluding zero number. After that, we have to take the difference between both. Thus we get the correct answer.