Question

What is the sum of all three dizit numbers whose all dizits are odd.

Answer 1

69375

Explanation:
The only odd digits are
1
,
3
,
5
,
7
,
9
, all of which are non-zero.

The number of ways of forming a three digit number from these digits is
5
3
=
125
, since there are
5
choices for the first digit,
5
for the second, and
5
for the third.

In these
125
ways, each digit has the same frequency.

The average digit value is
1
5
(
1
+
3
+
5
+
7
+
9
)
=
5
.

Each possible three digit number is a linear combination of digits.

Hence the average value of one of the three digit numbers is
555
.

So the sum is:
5
3
⋅
555
=
125
⋅
555
=
69375