6-digit passcode that contain exactly one multiple of 3 if digits can be repeated.

Question

Each element being any digits from 0 to 9. How many possible passcodes are there?

Tried doing $3 * 6 * 9 * 3 * 6 * 9$ but that's not exactly ONE multiple of 3.

Answer 1

I think I understand your question.

The multiples of $3$ are of course $3,6,9$. You can place exactly one of these in one of six slots. There are five remaining slots, with seven permissible digits for each: $0,1,2,4,5,7,8$.

So: three candidate "3 multiples" times 6 possible slots for that "multiple" times $7^5$ remaining "non-multiples."

If you count $0$ as a "3 multiple," then the answer changes in a straightforward manner:

$(4 \times 6) \times 6^5$.

Answer 2

One of the digits must be $0,3,6$ or $9$. The remaining $5$ must be $1,2,4,5,7,8$

You have $6$ choose for which digit is the multiple of three. And you have $4$ choices for which multiple of three it is. For the remaining five digits, you have $6$ choices which each one is.

So you have $6*4*6*6*6*6*6 = 4*6^5$ possibilities.