# five digit integer 3.4.5.6.7

#### Nekkamath

##### New member
Consider all of the positive five digit integers that can be formed using each of the digits 3,4,5,6,7 exactly once. What is the sum of these integers?

7 x 6 x 4
70 x 24
700 x 24
7000 x 24
70000x 24

I did this for each number (3,4,5,6) to come up with a grand total of 6,666,6000

#### soroban

##### Elite Member
Hello, Nekkamath!

Consider all of the positive five digit integers that can be formed
using each of the digits 3,4,5,6,7 exactly once.
What is the sum of these integers?

I approached it like this . . .

$$\displaystyle \text{Consider the list of }5! = 120\text{ permuations of those digits.}$$

. . $$\displaystyle \begin{array}{c}34567 \\ 34576 \\ 34657 \\ 34675 \\ 34756 \\ 34765 \\ \vdots \\ 76453 \\ 76534 \\76543 \end{array}$$

$$\displaystyle \text{In each column, each digit appears 24 times.}$$
. . $$\displaystyle \text{Hence, each column has a total of: }24 \times (3+4+5+6+7) \;=\;600$$

$$\displaystyle \text{The addition looks like this:}$$

. . . . . . . . . $$\displaystyle 6\:0\:0$$
. . . . . . . .$$\displaystyle 6\:0\:0$$
. . . . . . $$\displaystyle 6\:0\:0$$
. . . . .$$\displaystyle 6\:0\:0$$
. . . $$\displaystyle 6\:0\:0$$
. . .$$\displaystyle - - - - - -$$
. . . $$\displaystyle 6\:6\:6\:6\:6\:0\:0$$