Question on finding sum of all numbers formed by given numbers where zero is also included.

Question

Find the sum of all the odd numbers of five-digit that can be made using the digits $0,1,4,5,4.$

Answer 1

The number is a $5$ digit number. This means that $0$ cannot be the $1$st digit.

Also the number must be odd, so the last digit must be either $1$ or $5$. Equipped with this information we can proceed.

There are $2$ choices for the last digit.

There are $3$ choices for the first digit.

There are $3×2×1=6$ ways to arrange the remaining numbers.

$4$ is repeated so we have to divide by $2$

$$\frac {2\cdot 3 \cdot 6}{2}=18$$

Now for the sum of the numbers. $1$ is the last digit in half of the numbers and $5$ is the last digit in the other half. This makes the subtotal $9(1+5)=54$

Now when $1$ was the last digit, out of the $9$ numbers, $5$ was the first digit in exactly $3$ of them and $4$ was the first digit in exaclty $6$ of them. Repeating the same for the case when $5$ is the last digit makes the subtotal $(3(1+5)+12(4))\cdot 10000=660000$

Now lets look at the middle substring formed by the $2,3,4$th digits. This can be

$044,440,404$ each repeated $2$ times

$014,041,140,104,401,410$ each repeated once

$045,054,405,450,504,540$ each repeated once

The substrings above all need to be multiplied by $10$

Adding up all these subtotals, you can reach the answer