Question

The number of $5$ digit numbers such that the sum of their digit is even is :

• A. $50000$
• B. $45000$
• C. $60000$
• D. $noneofthese$

Answer 1

Answer: (B)

$45000$

The set of all $5$ digit numbers is $9\times {10}^4$ numbers strong.

We can partition this set into $9000$ subsets each of size $10$ such that in each subset, the first four digits are common and only the fifth one goes from $0$ to $9$.

Now, it is easy to see that exactly $5$ out of $10$ elements in each of these subsets has a digit sum which is even and since there are $900$ sum subsets means that there are $5\times 9000=45000$ numbers in the entire set that have an even digit sum.